MACS J0111.5+0855 | 1:11:27.73 | 08:55:28.6 | 1342237548 | 169 | single | 151 | 0.142 | 90218 | 22.39, 22.33
MACS J0455.2+0657 | 4:55:17.97 | 07:01:02.6 | 1342229655 | 169 | single | 433 | 0.097 | 90218 | 22.24, 22.13
MACS J0553.4–3342 | 5:53:27.79 | –33:42:35.1 | 1342227700 | 169 | multiple | 134 | 0.114 | 12005, 12123 | 23.14, 23.43
|  |  |  |  |  |  |  | 14281, 90218 |
MACS J0600.1–2008 | 6:00:23.90 | –20:06:38.0 | 1342230801 | 169 | single | 423 | 0.106 | 12005, 12123 | 23.10, 23.20
|  |  |  |  |  |  |  | 90218 |
RXC J0840.5+0544 | 8:40:32.09 | 05:45:01.1 | 1342230784 | 169 | single | 302 | 0.195 | 12095 | 22.21, 22.04
MACS J1115.8+0129 | 11:15:50.76 | 01:30:41.1 | 1342256866 | 1580 | single | 866 | 0.042 | 80168, 90213 | 23.66, 23.56
RXC J1314.3–2515 | 13:14:21.43 | –25:15:47.8 | 1342236193 | 169 | single | 1512 | 0.043 | 90218 | 22.24, 22.13
RXC J2104.8+1401 | 21:04:54.78 | 14:01:43.6 | 1342211300 | 169 | single | 302 | 0.078 | 90218 | 22.40, 22.45
RXC J2155.6+1231 | 21:55:41.33 | 12:31:50.8 | 1342211302 | 169 | multiple | 616 | 0.047 | 30344, 90218 | 22.18, 21.91
RXC J2332.4–5358ffHere we distinguish two portions of ALMA observations of R2332 obtained at different modes with slightly different spectral window settings. The multiple-pointing observation contains all the three components of this lensed SMGs with CO(7-6) coverage, and the single-pointing one targets pure dust continua of only two components. | 23:32:26.46 | –53:58:41.2 | 1342234736 | 169 | multiple | 605 | 0.088 | 40370, 60099 | 22.64, 21.84
|  |  |  |  |  |  |  | 60194, 80096 |
RXC J2332.4–5358ffHere we distinguish two portions of ALMA observations of R2332 obtained at different modes with slightly different spectral window settings. The multiple-pointing observation contains all the three components of this lensed SMGs with CO(7-6) coverage, and the single-pointing one targets pure dust continua of only two components. |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | single | 141 | 0.064 |  $\cdots$ |  $\cdots$

Table 2: Summary of Spitzer/IRAC, Herschel/SPIRE and ALMA Band-6 photometry

Galaxy ID | Short ID | CoordinatesaaALMA coordinates of detected source. |  Spitzer/IRAC |  Herschel/SPIRE | ALMA
---|---|---|---|---|---
|  | RA | Dec | [3.6 $\mathrm{\SIUnitSymbolMicro}$m] | [4.5 $\mathrm{\SIUnitSymbolMicro}$m] | $f_{250}$ | $f_{350}$ | $f_{500}$ | S/NpeakbbDefined as the maximum signal-to-noise ratio of peak flux density ($f_{\mathrm{peak}}/\sigma_{\mathrm{RMS}}$) among the four-level continuum images (Section 2.3). | $f_{1.3\,\mathrm{mm}}$
|  | (deg) | (deg) | (mag) | (mag) | (mJy) | (mJy) | (mJy) |  | (mJy)
HLS J004335.3–204203 | HLS0043-A | 10.89702 | –20.70082 | 18.94$\pm$0.02 | 18.90$\pm$0.07 | 81.6$\pm$8.1 | 70.3$\pm$7.7 | 61.2$\pm$7.3 | 9.9 | 2.50$\pm$0.50
HLS J004335.1–204159 | HLS0043-B | 10.89642 | –20.69959 | 20.93$\pm$0.11 | 20.08$\pm$0.07 | 20.7$\pm$5.0 |  $<$11.0 |  $<$11.0 | 4.7 | 2.69$\pm$0.56
HLS J011127.9+085525 | HLS0111-A | 17.86606 | 8.92353 | 21.66$\pm$0.06 | 20.98$\pm$0.05 | 27.3$\pm$5.4 | 36.0$\pm$5.7 | 19.5$\pm$4.4 | 19.4 | 1.98$\pm$0.40
HLS J011127.7+085529 | HLS0111-B | 17.86547 | 8.92467 | 19.71$\pm$0.02 | 19.37$\pm$0.02 | 45.1$\pm$6.3 | 59.4$\pm$7.1 | 32.2$\pm$5.3 | 19.1 | 3.26$\pm$0.33
HLS J011439.3–412404 | HLS0114 | 18.66373 | –41.40105 | 18.24$\pm$0.02 | 18.15$\pm$0.03 | 81.5$\pm$8.1 | 62.3$\pm$7.2 | 33.3$\pm$5.3 | 26.5 | 1.51$\pm$0.10
HLS J030707.7–284017 | HLS0307-28-A | 46.78209 | –28.67132 | 20.67$\pm$0.04 | 20.40$\pm$0.03 | 103.0$\pm$9.2 | 85.6$\pm$8.6 | 43.0$\pm$6.0 | 24.6 | 1.47$\pm$0.14
HLS J030708.2–284027 | HLS0307-28-B | 46.78405 | –28.67422 | 21.57$\pm$0.07 | 22.02$\pm$0.09 | 13.3$\pm$4.7 | 24.7$\pm$5.0 | 17.8$\pm$4.2 | 36.2 | 2.30$\pm$0.15
HLS J030749.6–504138 | HLS0307-50 | 46.95667 | –50.69391 | 19.99$\pm$0.06 | 19.40$\pm$0.04 | 102.1$\pm$9.1 | 107.1$\pm$9.9 | 67.2$\pm$7.7 | 28.7 | 3.77$\pm$0.21
HLS J045518.1+070102 | HLS0455 | 73.82525 | 7.01721 | 19.79$\pm$0.04∙ | 19.83$\pm$0.04∙ | 102.4$\pm$9.1 | 130.5$\pm$11.3 | 98.7$\pm$9.9 | 58.1 | 10.71$\pm$0.27
HLS J050532.9–614357 | HLS0505 | 76.38710 | –61.73249 | 20.74$\pm$0.10∙ | 20.73$\pm$0.12∙ | 101.5$\pm$9.1 | 103.3$\pm$9.7 | 76.3$\pm$8.3 | 32.1 | 4.38$\pm$0.30
HLS J054639.2–534520 | HLS0546 | 86.66346 | –53.75560 | 19.85$\pm$0.18 | 19.34$\pm$0.08 | 69.8$\pm$7.5 | 46.3$\pm$6.3 | 44.0$\pm$6.1 | 7.0 | 1.91$\pm$0.45
HLS J055327.8–334216 | HLS0553-A | 88.36579 | –33.70446 | 19.27$\pm$0.02 | 19.27$\pm$0.02 | 102.6$\pm$9.1 | 74.9$\pm$8.0 | 40.0$\pm$5.8 | 24.9 | 1.71$\pm$0.16
HLS J055327.6–334244 | HLS0553-B | 88.36510 | –33.71220 | 19.03$\pm$0.02 | 19.07$\pm$0.02 | 103.7$\pm$9.2 | 75.7$\pm$8.0 | 40.4$\pm$5.8 | 23.2 | 1.72$\pm$0.17
HLS J055327.8–334231 | HLS0553-C | 88.36602 | –33.70849 | 18.71$\pm$0.04 | 18.68$\pm$0.05 | 186.8$\pm$13.3 | 136.4$\pm$11.7 | 72.8$\pm$8.1 | 35.4 | 3.11$\pm$0.19
HLS J060023.8–200638 | HLS0600 | 90.09901 | –20.11067 | 18.57$\pm$0.03∙ | 18.29$\pm$0.03∙ | 173.7$\pm$12.7 | 270.5$\pm$19.7 | 237.8$\pm$19.6 | 77.0 | 34.27$\pm$0.61
HLS J061203.5–431712 | HLS0612 | 93.01468 | –43.28678 | 19.78$\pm$0.14 | 19.57$\pm$0.05 | 93.0$\pm$8.7 | 95.0$\pm$9.2 | 62.4$\pm$7.4 | 28.7 | 6.31$\pm$0.37
HLS J084032.1+054503 | HLS0840 | 130.13366 | 5.75073 | 18.94$\pm$0.03 | 18.78$\pm$0.03 | 90.5$\pm$8.5 | 85.5$\pm$8.6 | 41.0$\pm$5.9 | 8.0 | 4.00$\pm$0.90
HLS J111550.7+013036 | HLS1115 | 168.96117 | 1.50988 | 18.92$\pm$0.02 | 18.62$\pm$0.02 | 91.1$\pm$8.6 | 95.5$\pm$9.2 | 57.6$\pm$7.0 | 31.4 | 2.58$\pm$0.13
HLS J112402.2+240447 | HLS1124-A | 171.00896 | 24.07963 | 20.60$\pm$0.08∙ | 20.59$\pm$0.08∙ | 21.8$\pm$5.1 | 26.5$\pm$5.1 | 18.5$\pm$4.3 | 10.9 | 1.54$\pm$0.20
HLS J112402.3+240437 | HLS1124-B | 171.00967 | 24.07699 | 18.36$\pm$0.02∙ | 18.37$\pm$0.03∙ | 191.0$\pm$13.6 | 231.6$\pm$17.4 | 161.7$\pm$14.3 | 51.7 | 13.50$\pm$0.39
HLS J131421.3–251546 | HLS1314ccCombination of all the three clumps (HLS1314-A/B/C) seen in the ALMA map. | 198.58876 | –25.26276 | 16.84$\pm$0.01 | 16.74$\pm$0.01 | 130.2$\pm$10.5 | 141.2$\pm$12.0 | 93.1$\pm$9.5 | 26.5 | 4.58$\pm$0.21
HLS J162346.5+263412 | HLS1623 | 245.94394 | 26.57011 | 20.60$\pm$0.04 | 20.07$\pm$0.04 | 76.9$\pm$7.8 | 68.0$\pm$7.6 | 43.3$\pm$6.0 | 12.7 | 0.39$\pm$0.07
HLS J210454.6+140149 | HLS2104-A | 316.22738 | 14.03040 | 19.83$\pm$0.03 | 19.48$\pm$0.03 | 34.8$\pm$5.7 | 35.7$\pm$5.6 | 28.1$\pm$5.0 | 15.6 | 2.18$\pm$0.28
HLS J210454.7+140141 | HLS2104-B | 316.22777 | 14.02815 | 18.85$\pm$0.02 | 18.71$\pm$0.02 | 49.4$\pm$6.5 | 50.8$\pm$6.5 | 40.0$\pm$5.8 | 19.9 | 3.10$\pm$0.29
HLS J215540.5+123208 | HLS2155-A | 328.91861 | 12.53547 | 20.46$\pm$0.06 | 19.93$\pm$0.06 | 61.1$\pm$7.1 | 58.1$\pm$7.0 | 33.9$\pm$5.4 | 5.9 | 0.97$\pm$0.24
HLS J215540.8+123135 | HLS2155-B | 328.92018 | 12.52629 | 19.68$\pm$0.11 | 19.37$\pm$0.06 | 36.6$\pm$5.8 | 34.1$\pm$5.5 | 27.0$\pm$4.9 | 9.4 | 3.58$\pm$0.45
HLS J233227.2–535844 | HLS2332-A | 353.11334 | –53.97895 | 19.44$\pm$0.04 | 19.11$\pm$0.03 | 116.8$\pm$9.8 | 138.4$\pm$11.8 | 102.2$\pm$10.2 | 73.6 | 7.64$\pm$0.20
HLS J233225.5–535839 | HLS2332-B | 353.10645 | –53.97746 | 19.62$\pm$0.02 | 19.09$\pm$0.02 | 136.0$\pm$10.8 | 161.2$\pm$13.2 | 119.1$\pm$11.3 | 83.0 | 8.90$\pm$0.21
HLS J233229.5–535840 | HLS2332-C | 353.12295 | –53.97767 | 19.43$\pm$0.01 | 18.97$\pm$0.01 | 145.2$\pm$11.3 | 172.0$\pm$13.8 | 127.1$\pm$11.9 | 83.1 | 9.50$\pm$0.22

∙∙footnotetext: Highly-blended object with complex morphology. IRAC photometry
is obtained through a simultaneous MCMC fitting of ALMA and optical source
models (see descriptions in Section 3.2 and Figure 16).

Table 3: Summary of source structural parameters in ALMA Band 6 and
Spitzer/IRAC

ID | ALMA Band 6 | IRAC 3.6 $\mathrm{\SIUnitSymbolMicro}$m | IRAC 4.5 $\mathrm{\SIUnitSymbolMicro}$m | Offset
---|---|---|---|---
| $R_{\mathrm{e}}$ | $b/a$ | $R_{\mathrm{e}}$ | $b/a$ | $n$ | $R_{\mathrm{e}}$ | $b/a$ | $n$ |  $\Delta$RA |  $\Delta$Dec
| (′′) |  | (′′) |  |  | (′′) |  |  | (′′) | (′′)
HLS0043-A | 0.28$\pm$0.05 | 0.28$\pm$0.10 | 0.59$\pm$0.03 | 0.92$\pm$0.07 | 1.2$\pm$0.3 | 0.94$\pm$0.03 | 0.14$\pm$0.06 | 0.7$\pm$0.1 | –0.14 | –0.02
HLS0043-B | 0.27$\pm$0.09 | 0.41$\pm$0.25 |  $<$0.40 |  $\cdots$ |  $\cdots$ |  $<$0.40 |  $\cdots$ |  $\cdots$ | 0.01 | 0.01
HLS0111-A | 0.24$\pm$0.05 | 0.51$\pm$0.18 | 0.78$\pm$0.01 | 0.67$\pm$0.03 | [0.2] | 0.81$\pm$0.02 | 0.69$\pm$0.02 | 1.0$\pm$0.2 | 0.03 | –0.07
HLS0111-B | 0.13$\pm$0.03 | 0.29$\pm$0.34 | 0.48$\pm$0.13 | [0.54] | [0.2] | 0.45$\pm$0.07 | 0.56$\pm$0.30 | [0.2] | 0.06 | –0.16
HLS0114 | 0.24$\pm$0.02 | 0.44$\pm$0.08 | 0.53$\pm$0.03 | 0.36$\pm$0.06 | 1.5$\pm$0.4 | 0.69$\pm$0.03 | 0.82$\pm$0.05 | 1.0$\pm$0.3 | –0.04 | 0.08
HLS0307-28-A | 0.20$\pm$0.03 | 0.65$\pm$0.23 | 0.40$\pm$0.12 | 0.32$\pm$0.72 | [0.5] | 0.34$\pm$0.05 | 0.89$\pm$0.24 | [0.5] | 0.07 | –0.33
HLS0307-28-B | 0.15$\pm$0.03 | 0.09$\pm$1.37 |  $<$0.40 |  $\cdots$ |  $\cdots$ |  $<$0.40 |  $\cdots$ |  $\cdots$ | 0.10 | 0.01
HLS0307-50 | 0.27$\pm$0.03 | 0.30$\pm$0.06 | 0.64$\pm$0.07 | 0.35$\pm$0.21 | [0.2] | 0.97$\pm$0.05 | 0.39$\pm$0.07 | [0.2] | –0.14 | –0.06
HLS0546 | 1.68$\pm$0.27 | 0.99$\pm$0.26 | 1.16$\pm$0.26 | 0.05$\pm$0.21 | 1.8$\pm$1.1 | 1.02$\pm$0.21 | 0.04$\pm$0.12 | 2.8$\pm$1.0 | 1.20 | 0.14
HLS0553-A | 0.10$\pm$0.01 | 0.56$\pm$0.10 | 0.33$\pm$0.02 | 0.62$\pm$0.12 | [1.0] | 0.35$\pm$0.02 | 0.88$\pm$0.08 | [1.0] | 0.05 | –0.16
HLS0553-B | 0.15$\pm$0.01 | 0.43$\pm$0.05 | 0.76$\pm$0.01 | 0.32$\pm$0.03 | [1.0] | 0.85$\pm$0.03 | 0.54$\pm$0.02 | [1.0] | 0.09 | –0.36
HLS0553-C | 0.14$\pm$0.01 | 0.58$\pm$0.12 | 0.67$\pm$0.02 | 0.48$\pm$0.03 | [1.0] | 0.64$\pm$0.01 | 0.60$\pm$0.02 | [1.0] | 0.03 | –0.16
HLS0612 | 0.37$\pm$0.02 | 0.83$\pm$0.08 | 0.99$\pm$0.14 | 0.62$\pm$0.07 | 1.7$\pm$0.9 | 0.68$\pm$0.07 | 0.81$\pm$0.10 | 1.0$\pm$0.6 | –0.01 | 0.28
HLS0840 | 3.66$\pm$1.71 | 0.43$\pm$0.30 | 1.13$\pm$0.02 | 0.40$\pm$0.03 | [0.2] | 1.15$\pm$0.03 | 0.49$\pm$0.02 | 0.2$\pm$0.1 | 0.22 | 0.22
HLS1115 | 0.72$\pm$0.06 | 0.21$\pm$0.03 | 0.88$\pm$0.01 | 0.32$\pm$0.01 | 1.0$\pm$0.1 | 0.87$\pm$0.01 | 0.44$\pm$0.01 | 0.8$\pm$0.0 | –0.08 | –0.07
HLS1314-A | 1.07$\pm$0.40 | 0.27$\pm$0.07 | 3.72$\pm$0.10 | [0.16] | [0.2] | 3.42$\pm$0.13 | 0.14$\pm$0.01 | 0.2$\pm$0.1 | –0.10 | 0.08
HLS1314-B | 0.50$\pm$0.16 | 0.50$\pm$0.17 | 1.18$\pm$0.04 | [0.17] | 1.2$\pm$0.2 | 1.21$\pm$0.12 | 0.18$\pm$0.02 | 1.1$\pm$0.2 | –0.30 | –0.03
HLS1314-C | 0.94$\pm$0.18 | 0.22$\pm$0.18 | 1.45$\pm$0.04 | 0.21$\pm$0.03 | 0.9$\pm$0.1 | 1.80$\pm$0.04 | 0.21$\pm$0.02 | 0.8$\pm$0.1 | –0.12 | 0.32
HLS1623 | 0.89$\pm$0.32 | 0.81$\pm$0.04 | 0.17$\pm$0.11 | [0.71] | [0.2] | 0.77$\pm$0.03 | 0.71$\pm$0.06 | [0.2] | –0.05 | 0.72
HLS2104-A | 0.24$\pm$0.04 | 1.00$\pm$0.24 | 0.71$\pm$0.04 | 0.42$\pm$0.10 | [0.2] | 0.87$\pm$4.11 | 0.80$\pm$0.05 | 0.0$\pm$0.5 | –0.05 | 0.02
HLS2104-B | 0.43$\pm$0.05 | 0.58$\pm$0.10 | 1.13$\pm$0.02 | 0.75$\pm$0.02 | 0.7$\pm$0.1 | 1.08$\pm$0.02 | 0.73$\pm$0.02 | [1.0] | –0.00 | 0.12
HLS2155-A | 0.33$\pm$0.05 | 0.16$\pm$0.11 | 0.88$\pm$0.07 | [0.20] | [0.5] | 0.60$\pm$0.04 | [0.20] | [0.5] | 0.00 | –0.01
HLS2155-B | 1.52$\pm$0.10 | 0.47$\pm$0.05 | 2.01$\pm$0.26 | 0.41$\pm$0.05 | [1.5] | 2.13$\pm$0.57 | 0.30$\pm$0.03 | 2.5$\pm$0.9 | 0.16 | –0.01
HLS2332-A | 0.21$\pm$0.00 | 0.58$\pm$0.01 | 0.93$\pm$0.04 | 0.42$\pm$0.05 | 0.9$\pm$0.3 | 0.99$\pm$0.01 | 0.48$\pm$0.01 | [1.0] | –0.03 | –0.08
HLS2332-B | 0.21$\pm$0.01 | 0.59$\pm$0.03 | 0.71$\pm$0.11 | 0.50$\pm$0.10 | 0.2$\pm$0.5 | 0.71$\pm$0.01 | 0.72$\pm$0.01 | [1.0] | –0.10 | –0.08
HLS2332-C | 0.20$\pm$0.01 | 0.67$\pm$0.04 | 1.22$\pm$0.05 | 0.62$\pm$0.04 | [1.0] | 1.00$\pm$0.01 | 0.60$\pm$0.01 | [1.0] | –0.39 | –0.08

Note. — Values enclosed with square brackets are fixed during morphological
fitting (Section 3.2). $R_{\mathrm{e}}$ is the effective radius, $b/a$ is the
ratio between semi-minor and semi-major axis and $n$ is the Sérsic index.
Spatial offsets in the last two columns are measured between ALMA and IRAC
(average of 3.6 and 4.5 $\mathrm{\SIUnitSymbolMicro}$m) centroids (see
Appendix B).

Table 4: Summary of source physical properties derived from SED fitting  ID | RedshiftaaPhotometric redshifts are presented as the median of their likelihood distributions with 1$\sigma$ confidence range. | log($M_{\ast}$)bbThese quantities are not corrected for lensing magnification. | log(SFR)bbThese quantities are not corrected for lensing magnification. | log(sSFR) | log($L_{\mathrm{IR}}$)bbThese quantities are not corrected for lensing magnification.ccDefined as the total IR luminosity integrated from 8 to 1000 $\mathrm{\SIUnitSymbolMicro}$m. | log($M_{\mathrm{dust}}$)bbThese quantities are not corrected for lensing magnification. |  $T_{\mathrm{dust}}$ddModeled with MBB spectrum with fixed dust emissivity at $\beta=1.8$. | $A_{V}$
---|---|---|---|---|---|---|---|---
|  | ($\mu^{-1}\mathrm{M}_{\odot}$) | ($\mu^{-1}\mathrm{M}_{\odot}\mathrm{y}\mathrm{r}^{-1}$) | ($\mathrm{G}\mathrm{y}\mathrm{r}^{-1}$) | ($\mu^{-1}\mathrm{L}_{\odot}$) | ($\mu^{-1}\mathrm{M}_{\odot}$) | (K) | (mag)
HLS0043-A | 1.61${}^{+0.79}_{-0.44}$ | 11.77$\pm$0.35 | 2.73$\pm$0.48 | –0.01$\pm$0.58 | 12.87$\pm$0.39 | 9.30$\pm$0.39 | 29.3$\pm$7.0 | 2.3$\pm$1.0
HLS0043-B | 3.23${}^{+0.48}_{-0.59}$ | 11.69$\pm$0.29 | 2.66$\pm$0.43 | –0.02$\pm$0.64 | 12.78$\pm$0.23 | 9.01$\pm$0.23 | 32.3$\pm$5.0 | 2.2$\pm$0.7
HLS0111-A | 2.27 | 11.94$\pm$0.21 | 2.55$\pm$0.23 | –0.38$\pm$0.40 | 12.78$\pm$0.10 | 8.96$\pm$0.10 | 31.5$\pm$1.9 | 5.5$\pm$0.5
HLS0111-B | 2.27 | 11.96$\pm$0.26 | 2.84$\pm$0.19 | –0.12$\pm$0.42 | 12.98$\pm$0.10 | 9.20$\pm$0.10 | 31.5$\pm$1.2 | 2.9$\pm$0.5
HLS0114 | 1.49${}^{+0.43}_{-0.27}$ | 11.94$\pm$0.32 | 2.71$\pm$0.42 | –0.21$\pm$0.64 | 12.88$\pm$0.27 | 8.92$\pm$0.27 | 35.4$\pm$5.2 | 2.0$\pm$0.9
HLS0307-28-A | 1.19${}^{+0.32}_{-0.23}$ | 11.61$\pm$0.39 | 2.59$\pm$0.38 | –0.01$\pm$0.61 | 12.79$\pm$0.27 | 8.97$\pm$0.27 | 33.4$\pm$4.2 | 7.1$\pm$1.6
HLS0307-28-B | 2.65${}^{+0.54}_{-0.30}$ | 10.70$\pm$0.32 | 2.44$\pm$0.14 | 0.74$\pm$0.29 | 12.63$\pm$0.18 | 9.15$\pm$0.18 | 29.6$\pm$3.7 | 2.0$\pm$0.9
HLS0307-50 | 1.93${}^{+0.53}_{-0.39}$ | 12.25$\pm$0.33 | 2.86$\pm$0.38 | –0.38$\pm$0.55 | 13.13$\pm$0.25 | 9.34$\pm$0.25 | 33.5$\pm$5.3 | 5.0$\pm$0.9
HLS0455 | 2.93 | 11.82$\pm$0.21 | 3.43$\pm$0.06 | 0.62$\pm$0.28 | 13.57$\pm$0.06 | 9.62$\pm$0.06 | 34.4$\pm$0.6 | 2.2$\pm$0.4
HLS0505 | 1.27${}^{+1.09}_{-0.22}$ | 11.31$\pm$0.44 | 2.61$\pm$0.41 | 0.44$\pm$0.52 | 12.74$\pm$0.40 | 9.62$\pm$0.40 | 24.6$\pm$7.1 | 5.1$\pm$1.6
HLS0546 | 1.89${}^{+0.50}_{-0.42}$ | 11.85$\pm$0.41 | 2.74$\pm$0.36 | –0.09$\pm$0.65 | 12.93$\pm$0.26 | 9.02$\pm$0.26 | 33.9$\pm$5.8 | 3.5$\pm$1.1
HLS0553-A | 1.14 | 11.45$\pm$0.03 | 2.68$\pm$0.10 | 0.22$\pm$0.10 | 12.85$\pm$0.09 | 8.99$\pm$0.09 | 31.2$\pm$1.5 | 3.8$\pm$0.2
HLS0553-B | 1.14 | 11.54$\pm$0.07 | 2.66$\pm$0.15 | 0.07$\pm$0.17 | 12.82$\pm$0.14 | 9.00$\pm$0.14 | 31.1$\pm$1.5 | 3.7$\pm$0.3
HLS0553-C | 1.14 | 11.74$\pm$0.32 | 2.91$\pm$0.18 | 0.18$\pm$0.47 | 13.05$\pm$0.10 | 9.25$\pm$0.10 | 31.4$\pm$1.1 | 4.2$\pm$0.9
HLS0600 | 2.87 | 12.51$\pm$0.27 | 3.72$\pm$0.12 | 0.22$\pm$0.38 | 13.84$\pm$0.07 | 10.29$\pm$0.07 | 29.8$\pm$0.3 | 2.7$\pm$0.5
HLS0612 | 2.31${}^{+0.59}_{-0.55}$ | 11.76$\pm$0.40 | 3.11$\pm$0.29 | 0.37$\pm$0.48 | 13.23$\pm$0.27 | 9.53$\pm$0.27 | 31.8$\pm$5.5 | 3.0$\pm$0.9
HLS0840 | 1.97${}^{+0.52}_{-0.58}$ | 11.84$\pm$0.36 | 2.97$\pm$0.36 | 0.12$\pm$0.53 | 13.09$\pm$0.30 | 9.26$\pm$0.30 | 33.3$\pm$6.4 | 2.2$\pm$0.9
HLS1115 | 1.59${}^{+0.05}_{-0.01}$ | 12.15$\pm$0.01 | 2.83$\pm$0.01 | –0.32$\pm$0.04 | 12.92$\pm$0.03 | 9.36$\pm$0.03 | 31.2$\pm$0.6 | 3.1$\pm$0.1
HLS1124-A | 1.80 | 10.80$\pm$0.30 | 2.36$\pm$0.14 | 0.57$\pm$0.40 | 12.44$\pm$0.13 | 9.03$\pm$0.13 | 27.0$\pm$1.5 | 1.7$\pm$0.6
HLS1124-B | 1.80 | 11.70$\pm$0.30 | 3.27$\pm$0.12 | 0.57$\pm$0.40 | 13.35$\pm$0.12 | 9.98$\pm$0.12 | 27.0$\pm$0.4 | 1.5$\pm$0.4
HLS1314eeCombination of all the three clumps (HLS1314-A/B/C) seen in the ALMA map (See Section 4.1). | 1.45 | 12.73$\pm$0.06 | 2.44$\pm$0.21 | –1.32$\pm$0.20 | 12.94$\pm$0.07 | 9.50$\pm$0.07 | 28.4$\pm$0.7 | 2.3$\pm$0.1
HLS1623 | 0.96${}^{+0.17}_{-0.45}$ | 11.84$\pm$0.48 | 2.26$\pm$0.41 | –0.60$\pm$0.53 | 12.68$\pm$0.34 | 8.42$\pm$0.34 | 22.0$\pm$3.8 | 9.2$\pm$1.4
HLS2104-A | 2.17${}^{+0.71}_{-0.54}$ | 11.73$\pm$0.33 | 2.60$\pm$0.41 | –0.08$\pm$0.56 | 12.77$\pm$0.31 | 9.10$\pm$0.31 | 31.0$\pm$6.3 | 2.3$\pm$0.9
HLS2104-B | 2.13${}^{+0.62}_{-0.63}$ | 11.87$\pm$0.30 | 2.79$\pm$0.44 | –0.07$\pm$0.56 | 12.91$\pm$0.33 | 9.28$\pm$0.33 | 30.7$\pm$6.2 | 1.5$\pm$0.8
HLS2155-A | 1.72${}^{+0.29}_{-0.30}$ | 11.92$\pm$0.35 | 2.63$\pm$0.33 | –0.31$\pm$0.60 | 12.87$\pm$0.20 | 8.81$\pm$0.20 | 37.0$\pm$4.7 | 5.2$\pm$1.1
HLS2155-B | 2.66${}^{+0.94}_{-0.73}$ | 11.72$\pm$0.31 | 2.92$\pm$0.37 | 0.20$\pm$0.45 | 12.99$\pm$0.32 | 9.25$\pm$0.32 | 31.9$\pm$7.4 | 1.6$\pm$0.7
HLS2332-A | 2.73 | 11.91$\pm$0.09 | 3.61$\pm$0.12 | 0.72$\pm$0.17 | 13.64$\pm$0.12 | 9.44$\pm$0.12 | 36.7$\pm$0.7 | 3.0$\pm$0.2
HLS2332-B | 2.73 | 11.73$\pm$0.12 | 3.64$\pm$0.04 | 0.93$\pm$0.15 | 13.70$\pm$0.07 | 9.51$\pm$0.07 | 36.7$\pm$0.6 | 2.9$\pm$0.1
HLS2332-C | 2.73 | 12.49$\pm$0.24 | 3.51$\pm$0.13 | 0.03$\pm$0.35 | 13.64$\pm$0.07 | 9.53$\pm$0.07 | 36.7$\pm$0.6 | 3.8$\pm$0.4

## References

  * Abbott et al. (2018) Abbott, T. M. C., Abdalla, F. B., Allam, S., et al. 2018, ApJS, 239, 18
  * Agius et al. (2013) Agius, N. K., Sansom, A. E., Popescu, C. C., et al. 2013, MNRAS, 431, 1929
  * An et al. (2019) An, F. X., Simpson, J. M., Smail, I., et al. 2019, ApJ, 886, 48
  * Aravena et al. (2020) Aravena, M., Boogaard, L., Gónzalez-López, J., et al. 2020, arXiv e-prints, arXiv:2006.04284
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Baldwin et al. (1981) Baldwin, J. A., Phillips, M. M., & Terlevich, R. 1981, PASP, 93, 5
  * Barro et al. (2013) Barro, G., Faber, S. M., Pérez-González, P. G., et al. 2013, ApJ, 765, 104
  * Barro et al. (2016a) Barro, G., Kriek, M., Pérez-González, P. G., et al. 2016a, ApJ, 827, L32
  * Barro et al. (2016b) Barro, G., Faber, S. M., Dekel, A., et al. 2016b, ApJ, 820, 120
  * Battisti et al. (2019) Battisti, A. J., da Cunha, E., Grasha, K., et al. 2019, ApJ, 882, 61
  * Bertin (2006) Bertin, E. 2006, in Astronomical Society of the Pacific Conference Series, Vol. 351, Astronomical Data Analysis Software and Systems XV, ed. C. Gabriel, C. Arviset, D. Ponz, & S. Enrique, 112
  * Bertin & Arnouts (1996) Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393
  * Bleem et al. (2015) Bleem, L. E., Stalder, B., de Haan, T., et al. 2015, ApJS, 216, 27
  * Bothwell et al. (2013) Bothwell, M. S., Smail, I., Chapman, S. C., et al. 2013, MNRAS, 429, 3047
  * Bournaud et al. (2011) Bournaud, F., Chapon, D., Teyssier, R., et al. 2011, ApJ, 730, 4
  * Calura et al. (2017) Calura, F., Pozzi, F., Cresci, G., et al. 2017, MNRAS, 465, 54
  * Calzetti et al. (2000) Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682
  * Casey et al. (2014a) Casey, C. M., Narayanan, D., & Cooray, A. 2014a, Phys. Rep., 541, 45
  * Casey et al. (2014b) Casey, C. M., Scoville, N. Z., Sanders, D. B., et al. 2014b, ApJ, 796, 95
  * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763
  * Charlot & Fall (2000) Charlot, S., & Fall, S. M. 2000, ApJ, 539, 718
  * Cheale et al. (2019) Cheale, R. A., Geach, J. E., Edge, A. C., et al. 2019, MNRAS, 484, 2757
  * Chen et al. (2015) Chen, C.-C., Smail, I., Swinbank, A. M., et al. 2015, ApJ, 799, 194
  * Cochrane et al. (2019) Cochrane, R. K., Hayward, C. C., Anglés-Alcázar, D., et al. 2019, MNRAS, 488, 1779
  * Coe et al. (2019) Coe, D., Salmon, B., Bradač, M., et al. 2019, ApJ, 884, 85
  * Combes et al. (2012) Combes, F., Rex, M., Rawle, T. D., et al. 2012, A&A, 538, L4
  * da Cunha et al. (2008) da Cunha, E., Charlot, S., & Elbaz, D. 2008, MNRAS, 388, 1595
  * da Cunha et al. (2010) da Cunha, E., Eminian, C., Charlot, S., & Blaizot, J. 2010, MNRAS, 403, 1894
  * da Cunha et al. (2015) da Cunha, E., Walter, F., Smail, I. R., et al. 2015, ApJ, 806, 110
  * Davé et al. (2010) Davé, R., Finlator, K., Oppenheimer, B. D., et al. 2010, MNRAS, 404, 1355
  * Dekel & Birnboim (2006) Dekel, A., & Birnboim, Y. 2006, MNRAS, 368, 2
  * Dekel & Birnboim (2008) —. 2008, MNRAS, 383, 119
  * Dekel & Burkert (2014) Dekel, A., & Burkert, A. 2014, MNRAS, 438, 1870
  * Dekel & Silk (1986) Dekel, A., & Silk, J. 1986, ApJ, 303, 39
  * Di Matteo et al. (2005) Di Matteo, T., Springel, V., & Hernquist, L. 2005, Nature, 433, 604
  * Díaz-Santos et al. (2017) Díaz-Santos, T., Armus, L., Charmandaris, V., et al. 2017, ApJ, 846, 32
  * Dudzevičiūtė et al. (2020) Dudzevičiūtė, U., Smail, I., Swinbank, A. M., et al. 2020, MNRAS, 494, 3828
  * Dunlop et al. (2017) Dunlop, J. S., McLure, R. J., Biggs, A. D., et al. 2017, MNRAS, 466, 861
  * Ebeling et al. (2017) Ebeling, H., Qi, J., & Richard, J. 2017, MNRAS, 471, 3305
  * Egami et al. (2010) Egami, E., Rex, M., Rawle, T. D., et al. 2010, A&A, 518, L12
  * Elbaz et al. (2018) Elbaz, D., Leiton, R., Nagar, N., et al. 2018, A&A, 616, A110
  * Finoguenov et al. (2020) Finoguenov, A., Rykoff, E., Clerc, N., et al. 2020, A&A, 638, A114
  * Flewelling et al. (2016) Flewelling, H. A., Magnier, E. A., Chambers, K. C., et al. 2016, arXiv e-prints, arXiv:1612.05243
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Franco et al. (2018) Franco, M., Elbaz, D., Béthermin, M., et al. 2018, A&A, 620, A152
  * Fujimoto et al. (2017) Fujimoto, S., Ouchi, M., Shibuya, T., & Nagai, H. 2017, ApJ, 850, 83
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Geach et al. (2017) Geach, J. E., Dunlop, J. S., Halpern, M., et al. 2017, MNRAS, 465, 1789
  * Genzel et al. (2003) Genzel, R., Baker, A. J., Tacconi, L. J., et al. 2003, ApJ, 584, 633
  * Gobat et al. (2018) Gobat, R., Daddi, E., Magdis, G., et al. 2018, Nature Astronomy, 2, 239
  * Gómez-Guijarro et al. (2018) Gómez-Guijarro, C., Toft, S., Karim, A., et al. 2018, The Astrophysical Journal, 856, 121
  * Gonzaga & et al. (2012) Gonzaga, S., & et al. 2012, The DrizzlePac Handbook
  * González-López et al. (2017) González-López, J., Bauer, F. E., Romero-Cañizales, C., et al. 2017, A&A, 597, A41
  * González-López et al. (2020) González-López, J., Novak, M., Decarli, R., et al. 2020, arXiv e-prints, arXiv:2002.07199
  * Greve et al. (2012) Greve, T. R., Vieira, J. D., Weiß, A., et al. 2012, ApJ, 756, 101
  * Griffin et al. (2010) Griffin, M. J., Abergel, A., Abreu, A., et al. 2010, A&A, 518, L3
  * Gullberg et al. (2019) Gullberg, B., Smail, I., Swinbank, A. M., et al. 2019, MNRAS, 490, 4956
  * Hainline et al. (2011) Hainline, L. J., Blain, A. W., Smail, I., et al. 2011, ApJ, 740, 96
  * Hatsukade et al. (2018) Hatsukade, B., Kohno, K., Yamaguchi, Y., et al. 2018, PASJ, 70, 105
  * Hickox et al. (2012) Hickox, R. C., Wardlow, J. L., Smail, I., et al. 2012, MNRAS, 421, 284
  * Hodge & da Cunha (2020) Hodge, J. A., & da Cunha, E. 2020, arXiv e-prints, arXiv:2004.00934
  * Hodge et al. (2016) Hodge, J. A., Swinbank, A. M., Simpson, J. M., et al. 2016, ApJ, 833, 103
  * Hodge et al. (2019) Hodge, J. A., Smail, I., Walter, F., et al. 2019, ApJ, 876, 130
  * Hopkins et al. (2008) Hopkins, P. F., Hernquist, L., Cox, T. J., & Kereš, D. 2008, ApJS, 175, 356
  * Hora et al. (2012) Hora, J. L., Marengo, M., Park, R., et al. 2012, in Proc. SPIE, Vol. 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, 844239
  * Hunt et al. (2015) Hunt, L. K., Draine, B. T., Bianchi, S., et al. 2015, A&A, 576, A33
  * Ikarashi et al. (2015) Ikarashi, S., Ivison, R. J., Caputi, K. I., et al. 2015, ApJ, 810, 133
  * Ivison et al. (2010) Ivison, R. J., Smail, I., Papadopoulos, P. P., et al. 2010, MNRAS, 404, 198
  * Jiménez-Andrade et al. (2020) Jiménez-Andrade, E. F., Zavala, J. A., Magnelli, B., et al. 2020, ApJ, 890, 171
  * Kennicutt & Evans (2012) Kennicutt, R. C., & Evans, N. J. 2012, ARA&A, 50, 531
  * Lang et al. (2019) Lang, P., Schinnerer, E., Smail, I., et al. 2019, ApJ, 879, 54
  * Laporte et al. (2017) Laporte, N., Bauer, F. E., Troncoso-Iribarren, P., et al. 2017, A&A, 604, A132
  * Li et al. (2019) Li, Z., French, K. D., Zabludoff, A. I., & Ho, L. C. 2019, ApJ, 879, 131
  * Litke et al. (2019) Litke, K. C., Marrone, D. P., Spilker, J. S., et al. 2019, ApJ, 870, 80
  * Lotz et al. (2011) Lotz, J. M., Jonsson, P., Cox, T. J., et al. 2011, ApJ, 742, 103
  * Lotz et al. (2017) Lotz, J. M., Koekemoer, A., Coe, D., et al. 2017, ApJ, 837, 97
  * Lutz et al. (2016) Lutz, D., Berta, S., Contursi, A., et al. 2016, A&A, 591, A136
  * Madau & Dickinson (2014) Madau, P., & Dickinson, M. 2014, ARA&A, 52, 415
  * Man et al. (2016) Man, A. W. S., Zirm, A. W., & Toft, S. 2016, ApJ, 830, 89
  * Martí-Vidal et al. (2014) Martí-Vidal, I., Vlemmings, W. H. T., Muller, S., & Casey, S. 2014, A&A, 563, A136
  * Martis et al. (2019) Martis, N. S., Marchesini, D. M., Muzzin, A., et al. 2019, arXiv e-prints, arXiv:1907.08152
  * McAlpine et al. (2019) McAlpine, S., Smail, I., Bower, R. G., et al. 2019, MNRAS, 488, 2440
  * McMullin et al. (2007) McMullin, J. P., Waters, B., Schiebel, D., Young, W., & Golap, K. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 376, Astronomical Data Analysis Software and Systems XVI, ed. R. A. Shaw, F. Hill, & D. J. Bell, 127
  * Miettinen et al. (2017) Miettinen, O., Delvecchio, I., Smolčić, V., et al. 2017, A&A, 606, A17
  * Narayanan et al. (2010) Narayanan, D., Dey, A., Hayward, C. C., et al. 2010, MNRAS, 407, 1701
  * Neeleman et al. (2019) Neeleman, M., Bañados, E., Walter, F., et al. 2019, ApJ, 882, 10
  * Nelson et al. (2016) Nelson, E. J., van Dokkum, P. G., Förster Schreiber, N. M., et al. 2016, ApJ, 828, 27
  * Newman et al. (2015) Newman, A. B., Belli, S., & Ellis, R. S. 2015, ApJ, 813, L7
  * Newman et al. (2018) Newman, A. B., Belli, S., Ellis, R. S., & Patel, S. G. 2018, ApJ, 862, 125
  * Nguyen et al. (2010) Nguyen, H. T., Schulz, B., Levenson, L., et al. 2010, A&A, 518, L5
  * Oguri (2010) Oguri, M. 2010, PASJ, 62, 1017
  * Oke & Gunn (1983) Oke, J. B., & Gunn, J. E. 1983, ApJ, 266, 713
  * Oogi & Habe (2013) Oogi, T., & Habe, A. 2013, MNRAS, 428, 641
  * Ott (2010) Ott, S. 2010, in Astronomical Society of the Pacific Conference Series, Vol. 434, Astronomical Data Analysis Software and Systems XIX, ed. Y. Mizumoto, K.-I. Morita, & M. Ohishi, 139
  * Peng et al. (2010) Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2010, AJ, 139, 2097
  * Piazzo et al. (2015) Piazzo, L., Calzoletti, L., Faustini, F., et al. 2015, MNRAS, 447, 1471
  * Pilbratt et al. (2010) Pilbratt, G. L., Riedinger, J. R., Passvogel, T., et al. 2010, A&A, 518, L1
  * Poglitsch et al. (2010) Poglitsch, A., Waelkens, C., Geis, N., et al. 2010, A&A, 518, L2
  * Postman et al. (2012) Postman, M., Coe, D., Benítez, N., et al. 2012, ApJS, 199, 25
  * Price-Whelan et al. (2018) Price-Whelan, A. M., Sipőcz, B. M., Günther, H. M., et al. 2018, AJ, 156, 123
  * Puglisi et al. (2019) Puglisi, A., Daddi, E., Liu, D., et al. 2019, ApJ, 877, L23
  * Rawle et al. (2014) Rawle, T. D., Egami, E., Bussmann, R. S., et al. 2014, ApJ, 783, 59
  * Rawle et al. (2016) Rawle, T. D., Altieri, B., Egami, E., et al. 2016, MNRAS, 459, 1626
  * Rieke et al. (2009) Rieke, G. H., Alonso-Herrero, A., Weiner, B. J., et al. 2009, ApJ, 692, 556
  * Rowlands et al. (2015) Rowlands, K., Wild, V., Nesvadba, N., et al. 2015, MNRAS, 448, 258
  * Rowlands et al. (2012) Rowlands, K., Dunne, L., Maddox, S., et al. 2012, MNRAS, 419, 2545
  * Rujopakarn et al. (2011) Rujopakarn, W., Rieke, G. H., Eisenstein, D. J., & Juneau, S. 2011, ApJ, 726, 93
  * Rujopakarn et al. (2019) Rujopakarn, W., Daddi, E., Rieke, G. H., et al. 2019, ApJ, 882, 107
  * Sanders et al. (1988) Sanders, D. B., Soifer, B. T., Elias, J. H., et al. 1988, ApJ, 325, 74
  * Santini et al. (2010) Santini, P., Maiolino, R., Magnelli, B., et al. 2010, A&A, 518, L154
  * Schreiber et al. (2015) Schreiber, C., Pannella, M., Elbaz, D., et al. 2015, A&A, 575, A74
  * Scoville et al. (2016) Scoville, N., Sheth, K., Aussel, H., et al. 2016, ApJ, 820, 83
  * Simpson et al. (2014) Simpson, J. M., Swinbank, A. M., Smail, I., et al. 2014, ApJ, 788, 125
  * Simpson et al. (2015) Simpson, J. M., Smail, I., Swinbank, A. M., et al. 2015, ApJ, 799, 81
  * Simpson et al. (2017) —. 2017, ApJ, 839, 58
  * Skibba et al. (2011) Skibba, R. A., Engelbracht, C. W., Dale, D., et al. 2011, ApJ, 738, 89
  * Smail et al. (1997) Smail, I., Ivison, R. J., & Blain, A. W. 1997, ApJ, 490, L5
  * Speagle et al. (2014) Speagle, J. S., Steinhardt, C. L., Capak, P. L., & Silverman, J. D. 2014, ApJS, 214, 15
  * Spilker et al. (2019) Spilker, J. S., Bezanson, R., Weiner, B. J., Whitaker, K. E., & Williams, C. C. 2019, ApJ, 883, 81
  * Spilker et al. (2016) Spilker, J. S., Marrone, D. P., Aravena, M., et al. 2016, ApJ, 826, 112
  * Stach et al. (2019) Stach, S. M., Dudzevičiūtė, U., Smail, I., et al. 2019, MNRAS, 487, 4648
  * Strandet et al. (2016) Strandet, M. L., Weiss, A., Vieira, J. D., et al. 2016, ApJ, 822, 80
  * Suess et al. (2017) Suess, K. A., Bezanson, R., Spilker, J. S., et al. 2017, ApJ, 846, L14
  * Swinbank et al. (2010) Swinbank, A. M., Smail, I., Longmore, S., et al. 2010, Nature, 464, 733
  * Swinbank et al. (2014) Swinbank, A. M., Simpson, J. M., Smail, I., et al. 2014, MNRAS, 438, 1267
  * Symeonidis et al. (2013) Symeonidis, M., Vaccari, M., Berta, S., et al. 2013, MNRAS, 431, 2317
  * Tacchella et al. (2016) Tacchella, S., Dekel, A., Carollo, C. M., et al. 2016, MNRAS, 458, 242
  * Tacchella et al. (2015) Tacchella, S., Carollo, C. M., Renzini, A., et al. 2015, Science, 348, 314
  * Tacchella et al. (2018) Tacchella, S., Carollo, C. M., Förster Schreiber, N. M., et al. 2018, ApJ, 859, 56
  * Tacconi et al. (2008) Tacconi, L. J., Genzel, R., Smail, I., et al. 2008, ApJ, 680, 246
  * Tadaki et al. (2020) Tadaki, K.-i., Belli, S., Burkert, A., et al. 2020, arXiv e-prints, arXiv:2009.01976
  * Toft et al. (2014) Toft, S., Smolčić, V., Magnelli, B., et al. 2014, ApJ, 782, 68
  * van der Wel et al. (2012) van der Wel, A., Bell, E. F., Häussler, B., et al. 2012, ApJS, 203, 24
  * van der Wel et al. (2014) van der Wel, A., Franx, M., van Dokkum, P. G., et al. 2014, ApJ, 788, 28
  * van Dokkum (2005) van Dokkum, P. G. 2005, AJ, 130, 2647
  * van Dokkum et al. (2008) van Dokkum, P. G., Franx, M., Kriek, M., et al. 2008, ApJ, 677, L5
  * Whitaker et al. (2017) Whitaker, K. E., Pope, A., Cybulski, R., et al. 2017, ApJ, 850, 208
  * Yan et al. (2006) Yan, R., Newman, J. A., Faber, S. M., et al. 2006, ApJ, 648, 281
  * Zavala et al. (2015) Zavala, J. A., Yun, M. S., Aretxaga, I., et al. 2015, MNRAS, 452, 1140
  * Zitrin et al. (2015) Zitrin, A., Fabris, A., Merten, J., et al. 2015, ApJ, 801, 44
  * Zolotov et al. (2015) Zolotov, A., Dekel, A., Mandelker, N., et al. 2015, MNRAS, 450, 2327


# Complete Solution of the Lady in the Lake Scenario111 This paper is based on
work performed at the Air Force Research Laboratory (AFRL) Control Science
Center. DISTRIBUTION STATEMENT A. Approved for public release. Distribution is
unlimited. AFRL-2024-0127; Cleared 09 JAN 2024.

Alexander Von Moll  Meir Pachter

(December 2023)

###### Abstract

In the Lady in the Lake scenario, a mobile agent, $L$, is pitted against an
agent, $M$, who is constrained to move along the perimeter of a circle. $L$ is
assumed to begin inside the circle and wishes to escape to the perimeter with
some finite angular separation from $M$ at the perimeter. This scenario has,
in the past, been formulated as a zero-sum differential game wherein $L$ seeks
to maximize terminal separation and $M$ seeks to minimize it. Its solution is
well-known. However, there is a large portion of the state space for which the
canonical solution does not yield a unique equilibrium strategy. This paper
provides such a unique strategy by solving an auxiliary zero-sum differential
game. In the auxiliary differential game, $L$ seeks to reach a point opposite
of $M$ at a radius for which their maximum angular speeds are equal (i.e., the
antipodal point). $L$ wishes to minimize the time to reach this point while
$M$ wishes to maximize it. The solution of the auxiliary differential game is
comprised of a Focal Line, a Universal Line, and their tributaries. The Focal
Line tributaries’ equilibrium strategy for $L$ is semi-analytic, while the
Universal Line tributaries’ equilibrium strategy is obtained in closed form.

##  1 Introduction

The Lady in the Lake scenario involves a mobile agent, the Lady, denoted $L$,
swimming in a circular lake and another agent, the Man (or Monster), denoted
$M$, whose motion is constrained to the perimeter of the lake. $L$ seeks to
reach the perimeter with maximum angular separation from $M$ while the latter
seeks to minimize the angular separation. $L$’s swimming speed is less than
$M$’s running speed (otherwise the solution is relatively trivial), however,
upon reaching the shore, $L$ can run faster than $M$.

The scenario first appeared in a column in Scientific American by Martin
Gardner in 1965. This original problem description was later collected in a
book [1] and was also posted in a collection of Gardner’s writings [2]. Later,
the scenario was formulated as a zero-sum differential game and solved as an
example in [3]. Again, the scenario was included as an example in Başar and
Olsder’s book [4] and an analytical solution was provided therein. According
to [4] the scenario also appeared in the Russian translation of Isaacs’ book
[5]. Then the scenario was revisited in [6], although, instead of analysis and
geometry, numerical methods were used to approximate a solution (presumably
because an analytical solution already existed for comparison purposes). These
numerical methods were based upon viscosity solutions of the Hamilton-Jacobi-
Isaacs (HJI) partial differential equation.

More recently, the Lady in the Lake scenario has been reintroduced, in much
the same way as the original, in the magazine Quanta as a mathematical puzzle
[7, 8]. However, in its new incarnation, a twist has been added: in [7, Puzzle
2] the reader is asked to determine (essentially) the equilibrium escape time
which $L$ seeks to minimize and $M$ seeks to maximize when $L$ starts in the
center of the lake. The readers’ and author’s solutions account for the
possibility of $M$ changing direction in order to foil $L$’s strategy in an
effort to drive at the true equilibrium solution. However, a full differential
game treatment of this problem (as well as the more general scenario of any
starting position for $L$) has not yet been presented and is out of the scope
of the current paper. Nonetheless, [9] analyzed an easier variant of this
problem for which $L$ begins outside the lake and seeks to enter in minimum
time subject to keeping $θ$, the angle between $L$ and $M$, non-zero. There
are also several papers on the topic of evading a finite-range Turret whose
solutions resemble the original Lady in the Lake solution with a few added
subsolutions [10, 11, 12].

Although the solution to the original Lady in the Lake scenario have been
well-established, several open questions remain (and were mentioned in [4]).
These questions have to do with a particular point in the lake from which $L$
can guarantee its minimum terminal angular separation. This point is opposite
of $M$ at a radius from the lake’s center for which $L$ and $M$’s maximum
angular speeds are equal, henceforth, the antipodal point, or $E$. If $L$ were
to start under the equilibrium trajectory emanating from $E$ she’d do best by
first reaching $E$ and subsequently exiting the lake along the associated
equilibrium trajectory. Therefore, the following questions was raised:

> If $L$ starts at the lake center and knows $M$’s current action, show that
> $L$ will reach the antipodal point, $E$.  [[, p.
> 394,]paraphrased]basar1982chapter

Some natural extensions to this question then include:

  1. 1.

How long will it take for $L$ to reach $E$ (i.e., what is the equilibrium,
$\min\max$ time)?

  2. 2.

What if $L$ starts from general position (i.e., not just starting at the
center of the lake)?

This paper answers all of these questions, effectively completing the solution
of the Lady in the Lake differential game by providing a unique strategy for
the players in a large region of the state space for which the canonical
strategy is undefined.

A recent work [13] has sought to address very similar questions. There, the
focus is on finding the minimum time trajectory for $L$ in the region of the
state space where she has angular speed advantage over $M$ (which is only a
subset of the region for which the canonical strategy for the $\min\max$
terminal angle game is non-unique/undefined). Ultimately, the authors specify
a nonlinear program which utilizes a general numerical optimization solver to
obtain minimum time trajectory resulting in $L$ maneuvering to $E$. This paper
builds upon that work by providing a solution which is closed-form for part of
the state space and semi-analytic in the other part.

Following in the footsteps of [5, 3, 4], the methodology used within this
paper is based upon differential game theory. In general, obtaining solutions
to differential games is a difficult endeavour as it involves solving the HJI,
a technique which suffers from the curse of dimensionality [14]. For example,
the Homicidal Chauffeur Differential Game (HCDG) has only two states and two
parameters and yet its solution (or, at least, the bulk of it) was the subject
of a PhD dissertation [15] and a multitude of follow-on publications. This is,
in part, due to the abundance and variety of singularities present in its
solution [4]. Fortunately, the Lady in the Lake differential game has two
states and only one parameter (in its most reduced formulation) and its
solution is far simpler than that of the HCDG. As will be shown, the solution,
presented here, concerning the $\min\max$ time to reach the point $E$ contains
some singularities of its own. In particular, the solution contains a Focal
Line (FL) – a line which is, itself, an equilibrium trajectory that has
tributary equilibrium trajectories that enter tangentially (c.f., e.g., [16,
17]). Additionally, the solution also contains a Universal Line (UL) – a line
which, like the FL, is an equilibrium trajectory, but its tributaries do not
enter tangentially. The UL was introduced in the seminal work by Isaacs [5].

The remainder of this paper is summarized as follows. Section 2 contains a
rederivation of the classical Lady in the Lake results. Section 3 presents all
of the new results for the $\min\max$ time to reach $E$ differential game.
It’s broken down into a subsection on the FL and its tributaries, Section 3.1,
a subsection on the UL and its tributaries, Section 3.2, and a summary of the
complete solution. Lastly, the paper is concluded in Section 4. Regarding
notation, many symbols are reused in each section and subsection but are
typically defined in a specific way for that context. For example, the symbol
$\mathscr{H}$ is used to denote the Hamiltonian which is defined differently
in the classical formulation than it is in the $\min\max$ time formulation.

##  2 The Classical Lady in the Lake Scenario [4]

In this section, the solution given by Başar and Olsder in [4] is rederived in
detail for the sake of completeness. Consider the state space region

$\mathcal{R}=\left\\{(r,θ)\mid 0\leq r\leq 1,\ 0\leq θ\leq π\right\\}$

where $μ<1$ is the speed of $L$. Without loss of generality, the angular
position of $L$ w.r.t. $M$ is assumed to be in the range
$θ\in\left[0,π\right]$. The relative dynamics are

$\displaystyle\dot{r}$ $\displaystyle=μ\cos ψ,\qquad$ $\displaystyle r(0)$
$\displaystyle=r_{0},$ (1) $\displaystyle\dot{θ}$
$\displaystyle=\frac{μ}{r}\sin ψ-ω,\qquad$ $\displaystyle θ(0)$
$\displaystyle=θ_{0},\ 0\leq t\leq t_{f},$ (2)

where $(r_{0},\ θ_{0})\in\mathcal{R}$, $ψ\in\left[-\pi,\pi\right]$, and
$ω\in\left[-1,1\right]$ (all without loss of generality). The radius of the
lake is set to 1 (again, without loss of generality)222This reduction of the
parameter space to just the ratio of agent speeds, $μ$, can be accomplished
through a scaling of space and time.. The cost/payoff functional is

$J\left(r,θ,ψ(\cdot),ω(\cdot)\right)=Φ\left(r_{f},θ_{f}\right)=θ_{f},$ (3)

which $L$ wishes to maximize and $M$ wishes to minimize. The terminal surface
is given by

$ϕ(r,θ)=r-1=0$ (4)

The Value function, if it exists, gives the equilibrium cost/payoff of the
differential game

$V(r,θ)=\max_{ψ(\cdot)}\min_{ω(\cdot)}θ_{f}=\min_{ω(\cdot)}\max_{ψ(\cdot)}θ_{f}.$
(5)

We begin by forming the Hamiltonian

$\mathscr{H}=λ_{r}μ\cos ψ+λ_{θ}\left(\frac{μ}{r}\sin ψ-ω\right),$ (6)

where $λ_{r}$ and $λ_{θ}$ are state adjoint variables. The equilibrium state
adjoint dynamics are given by [18]

$\displaystyle\dot{λ}_{r}$ $\displaystyle=-\frac{\partial\mathscr{H}}{\partial
r}=λ_{θ}\frac{μ}{r^{2}}\sin ψ$ (7) $\displaystyle\dot{λ}_{θ}$
$\displaystyle=0.$ (8)

The last equality implies that $λ_{θ}(t)=λ_{θ}\forall
t\in\left[0,t_{f}\right]$, i.e., that $λ_{θ}$ is constant along the entire
equilibrium trajectory. At termination, the state adjoint variables must
satisfy [18]

$\displaystyle λ_{r_{f}}$ $\displaystyle=\frac{\partial Φ}{\partial
r_{f}}+ν\frac{\partial ϕ}{\partial r_{f}}=ν$ (9) $\displaystyle λ_{θ}$
$\displaystyle=λ_{θ_{f}}=\frac{\partial Φ}{\partial θ_{f}}+ν\frac{\partial
ϕ}{\partial θ_{f}}=1,$ (10)

where $ν$ is an additional adjoint variable. The equilibrium heading for $L$
must maximize the Hamiltonian, which implies

$\cos ψ^{*}=\frac{λ_{r}}{\sqrt{λ_{r}^{2}+\frac{1}{r^{2}}}},\qquad\sin
ψ^{*}=\frac{1}{r\sqrt{λ_{r}^{2}+\frac{1}{r^{2}}}}.$ (11)

Meanwhile, the equilibrium control for $M$ must minimize the Hamiltonian,
which implies

$ω^{*}=1.$ (12)

At termination, the Hamiltonian must satisfy

$\mathscr{H}_{f}=-\frac{\partial Φ}{\partial t_{f}}-ν\frac{\partial
ϕ}{\partial t_{f}}=0.$ (13)

Furthermore, since the system is time-autonomous and
$\frac{\partial\mathscr{H}}{\partial t}=0$ we have $\mathscr{H}=0\forall t$.

Evaluating Equation 6 at final time and substituting in the equilibrium
controls, Equations 11 and 12, and solving for $ν$ gives

$ν=\sqrt{\frac{1}{μ^{2}}-1}.$ (14)

Note that $ν$ must be positive in order for $\dot{r}_{f}$ to be positive,
which is necessary for $L$ to exit the lake. Repeating this step for general
time gives

$λ_{r}=\sqrt{\frac{1}{μ^{2}}-\frac{1}{r^{2}}}.$ (15)

Again, the negative case of the square root can be ruled out since heading
towards the center of the lake is never advantageous along the equilibrium
trajectory. Substituting Equation 15 into Equation 11 gives

$\cos ψ^{*}=\sqrt{1-\frac{μ^{2}}{r^{2}}},\qquad\sin ψ^{*}=\frac{μ}{r}.$ (16)

Since $\sin ψ^{*}=\frac{μ}{r}$, it must be the case that $r\geq μ$. That is,
the equilibrium control strategy for $L$ is only defined when $r\geq μ$. As
noted in [4], $L$’s strategy corresponds to heading away from the tangent of
the circle of radius $μ$ and results in a straight line in the non-rotating
Cartesian coordinate system.

Substituting the equilibrium control strategies, Equations 16 and 12, into the
dynamics, Equations 1 and 2, and dividing gives

$\displaystyle\frac{\mathrm{d}θ}{\mathrm{d}r}$
$\displaystyle=-\frac{1}{μ}\sqrt{1-\frac{μ^{2}}{r^{2}}}$
$\displaystyle\int_{θ_{0}}^{θ_{f}}\mathrm{d}θ$
$\displaystyle=-\frac{1}{μ}\int_{r_{0}}^{r_{f}}\sqrt{1-\frac{μ^{2}}{r^{2}}}\mathop{\mathrm{d}r}$
$\displaystyle θ_{f}-θ_{0}$
$\displaystyle=-\frac{1}{μ}\left[\sqrt{r_{f}^{2}-μ^{2}}-μ\cos^{-1}\left(\frac{μ}{r_{f}}\right)-\sqrt{r_{0}^{2}-μ^{2}}+μ\cos^{-1}\left(\frac{μ}{r_{0}}\right)\right].$
(17)

By setting $r_{f}=1$ in the above, the Value function is given by

$V(r,θ)=θ-\sqrt{\frac{1}{μ^{2}}-1}+\cos^{-1}μ+\sqrt{\frac{r^{2}}{μ^{2}}-1}-\cos^{-1}\left(\frac{μ}{r}\right).$
(18)

Define $θ_{T}=V(μ,π)$, i.e.,

$θ_{T}=π-\sqrt{\frac{1}{μ^{2}}-1}+\cos^{-1}μ$ (19)

Note that $L$ can only escape from the point $E=(μ,π)$ if $θ_{T}>0$ which
implies that $μ>μ_{\text{crit}}\approx 0.21723$. For the remainder of the
paper it is assumed that $L$’s speed is above this critical value.

Now, define the equilibrium trajectory which departs from $E$ and exits the
lake as $B$. Based on Equations 17 and 19, then,

$B(r)=π-\sqrt{\frac{r^{2}}{μ^{2}}-1}+\cos^{-1}\left(\frac{μ}{r}\right),\qquad
r\in\left[μ,1\right].$ (20)

If the state is such that $θ<B(r)$ then $θ_{f}<θ_{T}$ from Equations 17 and
19. Therefore, it would be better for $L$ to navigate to the point $E$ and
depart along $B$ in order to achieve $θ_{f}=θ_{T}$. Figure 1 shows the
equilibrium trajectories for the classical solution. Note the large blank area
of the state space for which no unique equilibrium trajectory exists and $L$
is prescribed to swim to the point $E$ and subsequently take the $B$
trajectory.

Figure 1: Equilibrium flowfield for the classical Lady in the Lake
differential game for $μ=0.3$.

The curve $B$ is a barrier surface (in the language of Isaacs [5]). That is,
neither agent can steer the state of the system towards or across the surface
on their opponent’s respective side. For example, if the state $(r,θ)$, is
below $B$, then $L$ cannot force the state onto $B$ (hence why she is
prescribed to swim to $E$ first). Let $\overrightarrow{n}$ be a vector that is
normal (pointing up and to the right) to the curve $B$,

$\overrightarrow{n}=\begin{bmatrix}-\frac{\mathrm{d}θ}{\mathrm{d}r}&1\end{bmatrix}^{\top}.$
(21)

A barrier curve is characterized by

$\min_{ω}\max_{ψ}\left\\{\overrightarrow{n}\cdot\begin{bmatrix}\dot{r}&\dot{θ}\end{bmatrix}^{\top}\right\\}=0.$
(22)

Expanding this condition gives

$\min_{ω}\max_{ψ}-\frac{\mathrm{d}θ}{\mathrm{d}r}μ\cos ψ+\frac{μ}{r}\sin
ψ-ω=0,$

which implies that the minimizing and maximizing controls are, respectively,
$ω=1$ and

$\cos
ψ=\frac{\frac{\mathrm{d}θ}{\mathrm{d}r}}{\sqrt{\left(\frac{\mathrm{d}θ}{\mathrm{d}r}\right)^{2}+\frac{1}{r^{2}}}},\qquad\sin
ψ=\frac{1}{r\sqrt{\left(\frac{\mathrm{d}θ}{\mathrm{d}r}\right)^{2}+\frac{1}{r^{2}}}}.$

Taking the derivative of Equation 20 and substituting into the above
expressions shows that Equation 22 is indeed satisfied. Furthermore, the
condition holds for any curve that is an additive constant w.r.t. $B$, hence
why there is no hope in $L$ being able to reach $B$ from below.

##  3 Min-Max Time to Reach the Antipodal Point

In this section, we wish to obtain unique trajectories in the region of the
state space below the barrier, $B$, that are optimal in some sense.
Specifically, we aim to populate this region with trajectories which reach the
point $E=(μ,π)$ such that the time spent getting there is in equilibrium
w.r.t. the two agents’ control strategies.

###  3.1 Focal Line

######  Proposition 1.

There is a Focal Line (FL) given by

$\mathcal{F}=\left\\{(r,θ)\mid 0<r\leq μ,\ θ=π\right\\},$ (23)

wherein $L$’s equilibrium control keeps the state of the state of the system
on the line $θ=π$ (i.e., she chooses the heading, $ψ$, s.t. $\dot{θ}=0$):

$\sin ψ_{FL}=\frac{r}{μ},$ (24)

and $M$’s equilibrium control is

$ω_{FL}=1.$ (25)

###### Proof.

Since $r<μ$, $L$ needs to increase $r\to μ$. The goal of $L$ is to reach the
point $E=(μ,π)$. Any deviation of $θ$ from $π$ will need to be recovered at
some point along the trajectory in order to end up at $E$. Also $L$’s relative
control authority over the $θ$ state is decreasing as she increases $r$. Thus
any deviation would be best dealt with earlier in the trajectory rather than
later. Taking this argument to the extreme: it is best for $L$ to keep $θ=π$
along the entire trajectory. Regarding $M$’s control, he has some
informational advantage in that, technically, $L$ must know his instantaneous
control input in order for her to implement her singular control. However, if,
for example, $M$ were to switch many times (thereby forcing $L$ to have to
guess and possibly be wrong many times) $L$ could instead choose $\sin ψ=0$
and head directly to $E$, arriving in a shorter time. In other words, $M$’s
efforts to exploit his informational advantage are, themselves, easily
exploitable. Hence, $M$ should adopt either $ω=1$ or $ω=-1$ while on the FL,
and thus the former is taken without loss of generality. ∎

######  Remark 1.

The proposed control for $L$ along the FL also keeps the state of the system
on the FL itself, which, of course, is one of the properties which makes this
surface a FL. The other property is that trajectories entering the FL do so
tangentially; this property will be proven later.

Substituting the FL controls, Equations 24 and 25, into the dynamics, Equation
1, gives

$\displaystyle\dot{r}$ $\displaystyle=μ\sqrt{1-\frac{r^{2}}{μ^{2}}}$
$\displaystyle=\sqrt{μ^{2}-r^{2}}.$

This expression can be used to obtain the amount of time spent on the FL until
the point $(r,θ)=(μ,π)$ is reached as follows.

$\displaystyle\dot{r}=\frac{\mathrm{d}r}{\mathrm{d}t}$
$\displaystyle=\sqrt{μ^{2}-r^{2}}$
$\displaystyle\frac{\mathrm{d}r}{\sqrt{μ^{2}-r^{2}}}$
$\displaystyle=\mathrm{d}t$
$\displaystyle\frac{\mathrm{d}r}{μ\sqrt{1-\frac{r^{2}}{μ^{2}}}}$
$\displaystyle=\mathrm{d}t$

Let $x\equiv\frac{r}{μ}$, and thus, $μ\mathrm{d}x=\mathrm{d}r$:

$\displaystyle\frac{\mathrm{d}x}{\sqrt{1-x^{2}}}$ $\displaystyle=\mathrm{d}t$

Finally, this equation can be integrated; on the LHS the integration bounds
are $x=\tfrac{s}{μ}$ to $1$ (which corresponds to $r$ starting at $s$ and
going to $μ$), and the RHS just becomes the time spent on the FL, $t_{s}$:

$\displaystyle\int_{s/μ}^{1}\frac{\mathrm{d}x}{\sqrt{1-x^{2}}}$
$\displaystyle=\int_{0}^{t_{s}}\mathrm{d}t=t_{s}$
$\displaystyle\implies\left.\sin^{-1}\left(x\right)\right|_{s/μ}^{1}$
$\displaystyle=t_{s}$
$\displaystyle\frac{π}{2}-\sin^{-1}\left(\frac{s}{μ}\right)$
$\displaystyle=t_{s}$

With the time spent on the FL in hand, the next step is to characterize the FL
tributaries, which are those equilibrium trajectories that merge onto the FL.
In order to do so, the game is reformulated as a game which begins from a
general initial condition and ends on the FL.

####  3.1.1 Equilibrium Heading for FL Tributaries

The terminal manifold is

$\mathcal{M}=\left\\{(r,θ)\mid 0<r\leq μ,\ θ=π\right\\}.$ (26)

$\mathcal{M}$ is also the zero-level set of the function

$ϕ(r,θ)=θ-π.$ (27)

For the remainder of the paper, $s$ is used to denote the value of $r$ wherein
the state enters the FL. Thus, for the analysis concerning FL tributaries
$r_{f}=s$ and $θ_{f}=π$ The terminal cost is the time for $L$ to proceed along
the line $θ=π$ from $r=s$ to $r=μ$:

$Φ(s)=\frac{π}{2}-\sin^{-1}\left(\frac{s}{μ}\right).$ (28)

The performance functional is the total time taken by $L$ to reach the point
$E\equiv\left(μ,π\right)$ (by way of reaching $(r,θ)=(s,π)$ first)

$J(ψ(\cdot))=Φ(s)+\int_{0}^{t_{f}}1\mathop{\mathrm{d}t},$ (29)

which $L$ wishes to minimize. The Hamiltonian of the system is

$\displaystyle\mathscr{H}$ $\displaystyle=λ_{r}μ\cos
ψ+λ_{θ}\left(\frac{μ}{r}\sin ψ-ω\right)+1,$ (30)

where $λ_{r}$ and $λ_{θ}$ are adjoint variables associated with the $r$ and
$θ$ states, respectively. The value of the Hamiltonian at terminal time is
given by [18]

$\mathscr{H}_{f}=-\frac{\partial Φ}{\partial t}-ν\frac{\partial ϕ}{\partial
t}=0.$ (31)

Since $\frac{\partial\mathscr{H}}{\partial t}=0$ and the system’s dynamics are
time-autonomous we have $\frac{\mathrm{d}\mathscr{H}}{\mathrm{d}t}=0$ and thus
$\mathscr{H}(t)=0\forall t\in\left[0,t_{f}\right]$. The value of the adjoint
variables at terminal time are given by [18]

$\displaystyle λ_{r_{f}}$ $\displaystyle=\frac{\partial Φ}{\partial
s}+ν\frac{\partial ϕ}{\partial s}=\frac{-1}{\sqrt{μ^{2}-r_{f}^{2}}}$ (32)
$\displaystyle λ_{θ_{f}}$ $\displaystyle=\frac{\partial Φ}{\partial
θ_{f}}+ν\frac{\partial ϕ}{\partial θ_{f}}=ν,$ (33)

where $ν$ is an additional adjoint variable. The optimal adjoint dynamics are
given by [18]

$\displaystyle\dot{λ}_{r}$ $\displaystyle=-\frac{\partial\mathscr{H}}{\partial
r}=-λ_{θ}\frac{μ}{r^{2}}\sin ψ^{*}$ (34) $\displaystyle\dot{λ}_{θ}$
$\displaystyle=-\frac{\partial\mathscr{H}}{\partial θ}=0.$ (35)

Since $\dot{λ}_{θ}=0$ we have that $λ_{θ}=ν\ \forall
t\in\left[0,t_{f}\right]$.

The equilibrium action for $L$ is one that minimizes the Hamiltonian,
$ψ^{*}=\operatorname*{\arg\min}_{ψ}\mathscr{H}$, and therefore the vector
$\begin{bmatrix}\cos ψ^{*}&\sin ψ^{*}\end{bmatrix}$ should be antiparallel
with the vector $\begin{bmatrix}λ_{r}&\frac{ν}{r}\end{bmatrix}$

$\cos ψ^{*}=\frac{-λ_{r}}{\sqrt{λ_{r}^{2}+\frac{ν^{2}}{r^{2}}}},\qquad\sin
ψ^{*}=\frac{-ν}{r\sqrt{λ_{r}^{2}+\frac{ν^{2}}{r^{2}}}}$ (36)

Similarly, the equilibrium action for $M$ is one that maximizes the
Hamiltonian, $ω^{*}=\operatorname*{\arg\max}_{ω}\mathscr{H}$, hence,

$ω^{*}=-\operatorname{sign}ν.$ (37)

Assuming $ω>0$, substituting the equilibrium controls into the Hamiltonian,
Equation 30, gives

$\displaystyle\mathscr{H}^{*}=-μ\sqrt{λ_{r}^{2}+\frac{ν^{2}}{r^{2}}}-ν+1$
$\displaystyle=0$ $\displaystyle\implies\sqrt{λ_{r}^{2}+\frac{ν^{2}}{r^{2}}}$
$\displaystyle=\frac{1-ν}{μ}.$ (38)

Substituting the terminal value of $λ_{r}$, Equation 32, and the terminal
value of $r$ (i.e., $s$) into the above and solving for $ν$ gives

$ν=\frac{-s^{2}}{μ^{2}-s^{2}},$ (39)

which is negative since $s<μ$ along the FL. Substitution into Equation 37
confirms that, indeed, $ω$ is positive.

Substituting Equation 36 and Equation 39 into Equation 30, evaluating at a
general time, and solving for $λ_{r}$ gives

$λ_{r}=\pm\frac{1}{μ^{2}-s^{2}}\sqrt{μ^{2}-\frac{s^{4}}{r^{2}}}.$ (40)

From Equation 32 it’s clear that, at terminal time, $λ_{r}<0$ which from
Equation 36 and Equation 1 implies that $\dot{r}_{f}>0$. Also, Equation 40
shows that $λ_{r}=0$ when $r=\frac{s^{2}}{μ}$. Therefore, it must be the case
that $λ_{r}$ (and, consequently, $\dot{r}$) changes sign once the system
passes through $r=\tfrac{r_{f}^{2}}{μ}$ since
$\operatorname{sign}(\dot{λ}_{r})=-\operatorname{sign}(\sin ψ^{*})=-1$ from
Equations 34, 36 and 39.

######  Lemma 1.

The equilibrium heading for $L$ along FL tributaries is given by

$\cos ψ^{*}=\pm\sqrt{1-\frac{s^{4}}{μ^{2}r^{2}}},\qquad\sin
ψ^{*}=\frac{s^{2}}{μr}.$ (41)

###### Proof.

Substitution of Equations 38 and 39 into Equation 36 gives the above
expressions. ∎

######  Lemma 2.

The equilibrium control for $M$ along FL tributaries is given by

$ω^{*}=1.$ (42)

###### Proof.

The result follows directly from Equations 37 and 39. ∎

######  Lemma 3.

The equilibrium FL tributary trajectory is a straight line in the global
Cartesian frame.

###### Proof.

The proof follows the same steps as the proof for Lemma 2 in [9] and is thus
omitted for brevity. ∎

######  Lemma 4.

The FL tributaries enter the FL tangentially.

###### Proof.

Evaluating $\dot{θ}$ from Equation 2 at terminal time (i.e., $r=s$) and
substituting in the equilibrium controls, Equations 41 and 42, gives

$\dot{θ}^{*}_{f}=\frac{μ}{s}\left(\frac{s^{2}}{μs}\right)-1=0.$

The FL, itself, is a line of constant $θ$, hence the result holds. ∎

####  3.1.2 Equilibrium Flowfield

Now, the equilibrium heading for $L$ can be substituted into $\dot{r}$,
Equation 1, to obtain $\dot{r}^{*}$. However, the sign of $\cos ψ^{*}$ is not
known directly except at final time (wherein $\cos ψ^{*},\ \dot{r}>0$).
Therefore, it is useful to consider the retrograde equation for $r$ (denoted
with a circle instead of a dot, i.e., $\mathring{r}=-\dot{r}$) in order for
the initial condition to be fully specified:

$\mathring{r}^{*}=\pm\frac{μ}{r}\sqrt{r^{2}-\frac{s^{4}}{μ^{2}}},\qquad
r(0)=s,\ \mathring{r}^{*}(0)<0.$ (43)

As mentioned previously, the sign of $\mathring{r}^{*}$ is governed by the
sign of $λ_{r}$ which starts (in retrograde time) negative and becomes
positive if $r$ reaches the value $\frac{s^{2}}{μ}$. Let the retrograde time
be denoted by $τ$ such that $τ=0$ corresponds to $t=t_{f}$. Rewriting the
above expression,

$\displaystyle\frac{\mathrm{d}r}{\mathrm{d}τ}$
$\displaystyle=\pm\frac{μ}{r}\sqrt{r^{2}-\frac{s^{4}}{μ^{2}}}$ (44)
$\displaystyle\int_{s}^{r}\frac{r}{\sqrt{r^{2}-\frac{s^{4}}{μ^{2}}}}\mathop{\mathrm{d}r}$
$\displaystyle=\pm\int_{0}^{τ}μ\mathop{\mathrm{d}τ}$ (45)
$\displaystyle\left.\sqrt{r^{2}-\frac{s^{4}}{μ^{2}}}\ \right\rvert_{s}^{r}$
$\displaystyle=\pm μτ$ (46)
$\displaystyle\sqrt{r^{2}-\frac{s^{4}}{μ^{2}}}-\sqrt{s^{2}-\frac{s^{4}}{μ^{2}}}$
$\displaystyle=\pm μτ$ (47) $\implies
r(τ)=\sqrt{s^{2}-2τs\sqrt{μ^{2}-s^{2}}+μ^{2}τ^{2}}$ (48)

Define the time when $r=\frac{s^{2}}{μ}$ (i.e., when $λ_{r}$ and
$\mathring{r}$ change sign) as $\bar{τ}$; this time is obtained by solving for
$τ$ in the negative version of Equation 47 with $r=\frac{s^{2}}{μ}$:

$\bar{τ}=\frac{s}{μ}\sqrt{1-\frac{s^{2}}{μ^{2}}}.$ (49)

Note that this time also corresponds to the time at which $L$ is is closest to
the center of the lake along the FL trajectory, i.e.,
$\min_{τ}r(τ)=r(\bar{τ})$.

Similarly, for $θ$, after substituting Equations 41 and 42 into Equation 2 and
changing to retrograde time we have

$\mathring{θ}^{*}=1-\frac{s^{2}}{r^{2}},\qquad θ(0)=π.$ (50)

Rewriting the above expression and substituting in $r(τ)$ from Equation 48,

$\displaystyle\frac{\mathrm{d}θ}{\mathrm{d}τ}$
$\displaystyle=1-\frac{s^{2}}{s^{2}-2τs\sqrt{μ^{2}-s^{2}}+μ^{2}τ^{2}}$ (51)
$\displaystyle\int_{π}^{θ}\mathrm{d}θ$
$\displaystyle=\int_{0}^{τ}\mathrm{d}τ-s^{2}\int_{0}^{τ}\frac{1}{s^{2}-2sτ\sqrt{μ^{2}-s^{2}}+μ^{2}τ^{2}}\mathop{\mathrm{d}τ}$
(52) $\displaystyle θ-π$
$\displaystyle=τ-\tan^{-1}\left(\frac{μ^{2}}{s^{2}}τ-\sqrt{\frac{μ^{2}}{s^{2}}-1}\right)-\tan^{-1}\left(\sqrt{\frac{μ^{2}}{s^{2}}-1}\right).$
(53)

The following is stated in order to summarize the results of this section.

######  Lemma 5.

The equilibrium flowfield for FL tributaries, parameterized by the entry point
on the FL, $s$, is given by

$\displaystyle r(τ;s)$
$\displaystyle=\sqrt{s^{2}-2τs\sqrt{μ^{2}-s^{2}}+μ^{2}τ^{2}},$ (54)
$\displaystyle θ(τ;s)$
$\displaystyle=π+τ-\tan^{-1}\left(\frac{μ^{2}}{s^{2}}τ-\sqrt{\frac{μ^{2}}{s^{2}}-1}\right)-\tan^{-1}\left(\sqrt{\frac{μ^{2}}{s^{2}}-1}\right).$

####  3.1.3 Computation of the FL Entry Point

The equilibrium flowfield expressions derived in the previous section are
useful for filling a region of the state space with equilibrium trajectories
by computing $(r(τ),θ(τ))$ starting from points along the FL. However,
starting (in forward time) from a general position $(r,θ)$, the equilibrium
heading of $L$ is unknown as it depends on $s$. This section describes the
process by which $s$ may be computed.

There are two possible cases depending on $L$’s initial condition: 1) $L$’s
equilibrium heading has some component of towards the center of the lake and
2) $L$’s equilibrium heading has a component of velocity away from the center
of the lake until she reaches the FL. Consider Case 1. Let the time of arrival
of $L$ to the entry point of the FL, $(s,π)$ be

$t_{L}(s)=\frac{1}{μ}\left(\sqrt{r^{2}-\frac{s^{4}}{μ^{2}}}+\sqrt{s^{2}-\frac{s^{4}}{μ^{2}}}\right),$
(55)

which is derived based on the fact that $L$’s trajectory is a straight line in
the Cartesian frame (per Lemma 3) and is tangent to a circle of radius
$\frac{r_{f}^{2}}{μ}$. $M$’s time of arrival to the position that is antipodal
to $L$ is given by the sum of angles traversed

$t_{M}(s)=θ+\cos^{-1}\left(\frac{s^{2}}{μr}\right)+\cos^{-1}\left(\frac{s}{μ}\right)-π.$
(56)

Define the function $δ(s)=t_{L}(s)-t_{M}(s)$ which is the difference of the
agents’ respective times of arrival. The equilibrium entry point onto the FL
is thus the smallest possible root of this function, i.e.,

$s^{*}=\min s\qquad\text{ s.t. }δ(s)=0,\ s\in\left(0,\ μ\right].$ (57)

The solution may be obtained numerically as the above expression does not
admit a closed-form solution.

Case 2 is similar to Case 1 but with $t_{L}$ and $t_{M}$ given, respectively,
by

$\displaystyle t_{L}(s)$
$\displaystyle=\frac{1}{μ}\left(\sqrt{s^{2}-\frac{s^{4}}{μ^{2}}}-\sqrt{r^{2}-\frac{s^{4}}{μ^{2}}}\right),$
(58) $\displaystyle t_{M}(s)$
$\displaystyle=θ-\cos^{-1}\left(\frac{s^{2}}{μr}\right)+\cos^{-1}\left(\frac{s}{μ}\right)-π.$
(59)

In lieu of a more sophisticated method with which to determine whether the
initial condition, $(r,θ)$, is in Case 1 or Case 2, the former should be
assumed first. If no solution to Equation 57 can be found, then Case 2 should
be assumed.

###  3.2 Universal Line

######  Proposition 2.

There is a Universal Line (UL) given by

$\mathcal{U}=\left\\{(r,θ)~{}\mid~{}0\leq r\leq 1,\ θ=0\right\\},$ (60)

wherein $L$’s equilibrium control strategy is to head directly to the center
of the lake and $M$ does not move, i.e.,

$\cos ψ_{UL}=-1,\qquad ω_{UL}=0.$ (61)

###### Proof.

When $θ=0$, $M$ has no incentive to move the state of the system to some non-
zero $θ$ since doing so increases $L$’s angular separation (which is,
ultimately, the thing that $M$ seeks to reduce). If $L$ had an angular
component of velocity then $θ$ would immediately become non-zero. When $θ=0$,
the easiest way for $L$ to drive $θ\to π$ is to pass through the origin. ∎

Just as in the section on obtaining equilibrium controls for FL tributaries,
the game is reformulated as a game which begins from a general initial
condition and ends on the UL.

####  3.2.1 Equilibrium Heading for UL Tributaries

The terminal manifold is the set of states where $θ=0$, i.e.,

$\mathcal{M}=\left\\{(r,θ)\mid 0<r\leq 1,\ θ=0\right\\},$ (62)

which is also the zero-level set of the function

$ϕ(r,θ)=θ.$ (63)

The terminal cost is the time for $L$ to reach the origin along the UL under
the proposed UL strategy, Equation 61:

$Φ(r_{f},θ_{f})=\frac{r_{f}}{μ}.$ (64)

In principle, one may consider the total time to finish out the original game
by adding in the time spent along the FL, starting from $(0,π)$ and going to
$(μ,π)$, however that is not necessary as that time will be the same for all
UL tributaries. The performance functional is the sum of the time taken to
reach the UL and then reach the origin (i.e., Equation 29). The Hamiltonian is
the same as in Equation 30. Similarly as before, the equilibrium Hamiltonian
is zero for all time, and the equilibrium heading is given by Equation 36
resulting in Equation 38. The terminal adjoint values are

$\displaystyle λ_{r_{f}}$ $\displaystyle=\frac{\partial Φ}{\partial
r_{f}}+ν\frac{\partial ϕ}{\partial r_{f}}=\frac{1}{μ}$ (65) $\displaystyle
λ_{θ_{f}}$ $\displaystyle=\frac{\partial Φ}{\partial θ_{f}}+ν\frac{\partial
ϕ}{\partial θ_{f}}=ν.$ (66)

Evaluating Equation 38 at final time results in

$\sqrt{\frac{1}{μ^{2}}+\frac{ν^{2}}{r_{f}^{2}}}=\frac{1-ν}{μ}.$ (67)

Solving this expression, algebraically, for $ν$ yields
$ν=\frac{2r_{f}^{2}}{r_{f}^{2}-μ^{2}}$ which goes to infinity as $r_{f}\to μ$;
additionally the sign of $ν$ changes depending on whether $r_{f}\gtrless μ$.
Fortunately, the solution $ν=0$ is valid for all $r_{f},μ\in\left[0,1\right]$.

######  Lemma 6.

The equilibrium heading for $L$ along UL tributaries is given by

$\cos ψ=-1.$ (68)

###### Proof.

The result follows from the preceding analysis. Ultimately, $L$ must end at
the center of the lake and thus the $θ$ state bears no importance while $L$ is
en route. Therefore, the fastest way for $L$ to reach the center of the lake
is a straight line path, which is achieved with $ψ=π$. ∎

Note that, since $ν=0$, $M$’s control disappears from the Hamiltonian in
Equation 30 and therefore every value $ω\in\left[-1,1\right]$ is equally
optimal.

####  3.2.2 Equilibrium Flowfield

In contrast to the FL tributaries, the flowfield for the UL tributaries is
simple. Since $M$’s equilibrium control is undefined on the UL tributaries, we
adopt a value of $ω^{*}=1$.

######  Lemma 7.

The equilibrium flowfield for UL tributaries is given by

$\displaystyle r(τ)$ $\displaystyle=μτ,\qquad$ $\displaystyle
r(0)\in\left[0,1\right)$ (69) $\displaystyle θ(τ)$ $\displaystyle=τ,\qquad$
$\displaystyle θ(0)=0.$

A direct result of Lemma 7 is that UL tributaries only exist when
$θ\leq\frac{r}{μ}$. The interpretation is that UL tributaries exist when $M$
is close enough to $L$ so as to be able to close their angular separation
prior to the latter reaching the center of the lake.

###  3.3 Full Solution

The following result pieces together the two types of trajectories covered in
the previous section.

######  Lemma 8.

The line segment

$\mathcal{P}=\left\\{(r,θ)~{}\mid~{}0\leq r\leq 1,\ θ=\frac{r}{μ}\right\\}$
(70)

partitions the state space into two regions: one where FL tributaries exist
and are optimal and one where UL tributaries exist and are optimal. That is,
the two regions are mutually exclusive.

###### Proof.

It was shown previously, in Lemma 7, that UL tributaries exist below
$\mathcal{P}$. The remainder of the proof focuses on showing that FL
tributaries exist above $\mathcal{P}$, that is, for $θ>\frac{μ}{r}$. Consider
the FL tributary for which $r_{f}\to 0$; this is the most limiting case for FL
tributaries as the other endpoint of $\mathcal{F}$ (where $r=μ$) corresponds
to already being at the desired point (i.e., the trajectory is the single
point $(r,θ)=(μ,π)$). From Equations 43 and 50 we have

$\displaystyle\lim_{r_{f}\to 0}\mathring{r}^{*}\Big{|}_{r>0}$
$\displaystyle=+\frac{μ}{r}\sqrt{r^{2}-\frac{0^{4}}{μ}}=μ$
$\displaystyle\lim_{r_{f}\to 0}\mathring{θ}^{*}\Big{|}_{r>0}$
$\displaystyle=1-\frac{0^{2}}{r^{2}}=1$

These retrograde dynamics result in a line that is parallel to the partition
$\mathcal{P}$ and lies arbitrarily close to it since $0<r_{f}\ll 1$. Two
remaining properties are needed in order for the result to hold: 1) that the
FL tributaries do not cross one another (and thus no FL tributary crosses
below $\mathcal{P}$ as a result of the above analysis), and 2) that the FL
tributaries fill the region of the state space above $\mathcal{P}$. Both of
these properties will be verified, graphically, with an example. ∎

Based on all of the preceding results of this section, the following theorem
summarizes the solution of the min-max time game.

######  Theorem 1.

The solution to the zero-sum differential game of time to reach the antipodal
point $E$ is given by the following equilibrium control strategies and
associated Value function.

$\displaystyle\left(\cos ψ^{*},\ \sin ψ^{*}\right)$
$\displaystyle=\begin{cases}\left(\sqrt{1-\frac{r^{2}}{μ^{2}}},\
\frac{r}{μ}\right)&\text{ if }θ=π,\\\ \left(-1,\ 0\right)&\text{ if
}θ\leq\frac{r}{μ},\\\ \left(\pm\sqrt{1-\frac{r_{f}^{4}}{μ^{2}r^{2}}},\
\frac{r_{f}^{2}}{μr}\right)&\text{ otherwise. }\end{cases}$ (71)
$\displaystyle ω^{*}$ $\displaystyle=\begin{cases}1&\text{ if
}θ>\frac{r}{μ}\\\ 0&\text{ if }θ=0\\\ \text{undef.}&\text{ otherwise,
}\end{cases}$ (72) $\displaystyle t_{f}^{*}$
$\displaystyle=\begin{cases}\frac{π}{2}-\sin^{-1}\left(\frac{r}{μ}\right)&\text{
if }θ=π,\\\ \frac{π}{2}+\frac{r}{μ}&\text{ if }θ\leq\frac{r}{μ},\\\
\frac{π}{2}-\sin^{-1}\left(\frac{s}{μ}\right)+t_{L}(s),&\text{ otherwise,
}\end{cases}$ (73)

where $s$ is the solution of Equation 57 and $t_{L}(s)$ is given by Equation
55 or Equation 58 depending on which case applies to the current state as
described in Section 3.1.3. Note that the corresponding case determines the
sign of $\cos ψ^{*}$ as well.

######  Remark 2.

One may verify that the equilibrium control strategies satisfy the Hamilton-
Jacobi-Isaacs (HJI) equation everywhere via direct substitution. However, this
is true by construction since, in this case, the Hamiltonian is equivalent to
the HJI and the control strategies are derived directly from the former.

Figure 2 shows the relative state space filled with equilibrium trajectories.
Solutions to the classical game (i.e., the min-max angular separation when $L$
reaches $r=1$) exist above the barrier, $B$. It is assumed that $L$ would
utilize the classical strategy to exit the lake, otherwise, she should swim to
$E$ as quickly as possible and then exit the lake along the barrier, $B$.

Figure 2: Equilibrium trajectories of the complete Lady in the Lake game with
$μ=0.3$.

(a) Case 1 (b) Case 2

Figure 3:  Focal Line trajectories starting from the tributaries in the non-
rotating Cartesian coordinate system. In (a), $L$ initially heads towards the
tangent of the circle of radius $\frac{s^{2}}{μ}$ (Case 1), while in (b) $L$
only heads away from the tangent (Case 2). Open markers indicate initial
positions, triangles designate positions at the moment the FL is reached, and
closed markers indicate terminal positions.

##  4 Conclusion

Although the classical Lady in the Lake scenario has been solved for quite
some time, the question of what, specifically, to do “under” the barrier curve
was open. This paper has addressed that question by providing the $\min\max$
time and associated equilibrium strategies for $L$ to reach the antipodal
point. Subsequent to reaching the antipodal point, $L$ then continues on to
reach the shore and obtain the equilibrium terminal angular separation.
Traditional differential game theory methods have been used to obtain the
solution of the $\min\max$ time to reach the antipodal point game.
Interestingly, its solution is made up of two singular surfaces and their
tributaries. The approach taken in this paper will serve as a stepping stone
to address the more difficult game of $\min\max$ time to escape (i.e., similar
to the problem posed in [7]).

## References

  * [1] Martin Gardner  “Mathematical carnival - from penny puzzles, card shuffles and tricks of lightning calculators to roller coaster rides into the fourth dimension”  Knopf, 1975
  * [2] Martin Gardner  “Lady in Lake (1965)” box 15, folder 11  In _Martin Gardner Papers (SC0647)_ Stanford, California: Dept. of Special CollectionsUniversity Archives, Stanford University Libraries, 2008
  * [3] John V. Breakwell  “Lecture notes”  In _Differential Games and Applications_ Berlin, Heidelberg: Springer Berlin Heidelberg, 1977, pp. 70–95
  * [4] Tamer Başar and Geert Jan Olsder  “Chapter 8: Pursuit-Evasion Games”  In _Dynamic Noncooperative Game Theory_ 160, Mathematics in Science and Engineering  London: Elsevier, 1982, pp. 344–398  DOI: 10.1016/S0076-5392(08)62960-4
  * [5] Rufus Isaacs  “Differential Games: A Mathematical Theory with Applications to Optimization, Control and Warfare”  New York: Wiley, 1965
  * [6] M. Falcone  “Numerical methods for differential games based on partial differential equations”  In _International Game Theory Review_ 08, 2006, pp. 231–272  DOI: 10.1142/S0219198906000886
  * [7] Pradeep Mutalik  “Can Math help you escape a hungry bear?”  In _Quanta Magazine_ , 2021
  * [8] Pradeep Mutalik  “Math can, in theory, help you escape a hungry bear”  In _Quanta Magazine_ , 2021
  * [9] Alexander Von Moll, Meir Pachter, Daigo Shishika and Zachariah Fuchs  “Circular Target Defense Differential Games”  In _Transactions on Automatic Control_ 68 New York: IEEE, 2022, pp. 4065–4078  DOI: 10.1109/TAC.2022.3203357
  * [10] MN Ivanov and EP Maslov  “A problem of avoidance of a rotating segment”  In _Computers & Mathematics with Applications_ 26.6 Pergamon, 1993, pp. 67–75  DOI: 10.1016/0898-1221(93)90118-F
  * [11] A.. Galyaev and E.. Maslov  “Evading a rotating detection zone on a plane”  In _Journal of Computer and Systems Sciences International_ 52 Pleiades Publishing Ltd, 2013, pp. 377–385  DOI: 10.1134/s1064230713030076
  * [12] Alexander Von Moll et al.  “Turret Escape Differential Game” Presented at the 19th ISDG.  In _Journal of Dynamics and Games_ New York: American Institute of Mathematical Sciences, 2023  DOI: 10.3934/jdg.2023012
  * [13] Yancheng Wang  “Solving the Lady in the Lake Problem and its Fastest Optimal Strategy” Bachelor’s Thesis, 2022
  * [14] Pierre Bernhard  “Pursuit-evasion games and zero-sum two-person differential games”  In _Encyclopaedia of Systems and Control_ Springer, 2014, pp. 1103–1109
  * [15] Antony W. Merz  “The Homicidal Chauffeur - A Differential Game”, 1971
  * [16] Arik Melikyan and Pierre Bernhard  “Geometry of Optimal Paths around Focal Singular Surfaces in Differential Games”  In _Applied Mathematics and Optimization_ 52.1, 2005, pp. 23–37  DOI: 10.1007/s00245-004-0816-8
  * [17] J.. Breakwell and P. Bernhard  “A simple game with a singular focal line”  In _Journal of Optimization Theory and Applications_ 64 Springer ScienceBusiness Media LLC, 1990, pp. 419–428  DOI: 10.1007/bf00939457
  * [18] Arthur Earl Bryson and Yu-Chi Ho  “Applied Optimal Control: Optimization, Estimation and Control”  New York, USA: TaylorFrancis Group, 1975  DOI: 10.1201/9781315137667