r}_{\widehat{\mathbb{V}}}(t,x,y)|^{2}=\inf_{(\tilde{x},\tilde{y})\in\widehat{\mathbb{V}}_{b}(t)}\big{[}|x-\tilde{x}|^{2}+|y-\tilde{y}|^{2}\big{]}=\inf_{\tilde{x}\in\mathbb{R}^{d}}\inf_{\tilde{y}\in\mathbb{V}_{b}(t,\tilde{x})}\big{[}|x-\tilde{x}|^{2}+|y-\tilde{y}|^{2}\big{]}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad\qquad\quad~{}=\inf_{\tilde{x}\in\mathbb{R}^{d}}\big{[}|x-\tilde{x}|^{2}+|{\bf
r}_{\mathbb{V}}(t,\tilde{x},y)|^{2}\big{]}\leq|{\bf
r}_{\mathbb{V}}(t,x,y)|^{2}\end{array}\right.$ (7.29)

###### Example 7.4

Set $d=m=1$ and consider time invariant set values:

$\displaystyle\mathbb{V}(x)=[x-1,x+1]\subset\mathbb{R},\quad\widehat{\mathbb{V}}:=\\{(x,y):x\in\mathbb{R},y\in[x-1,x+1]\\}\subset\mathbb{R}^{2}.$

Clearly $\mathbb{V}_{b}(x)=\\{x-1,x+1\\}$. One can easily verify that,

$\displaystyle\displaystyle{\bf
r}_{\mathbb{V}}(x,y)=y-(x+1),\quad\mbox{for}~{}y\approx
x+1;\quad\mbox{and}\quad{\bf
r}_{\mathbb{V}}(x,y)=(x-1)-y,\quad\mbox{for}~{}y\approx x-1;$
$\displaystyle\displaystyle{\bf
r}_{\widehat{\mathbb{V}}}(x,y)={y-(x+1)\over\sqrt{2}},\quad\mbox{for}~{}y\approx
x+1;\quad\mbox{and}\quad{\bf
r}_{\widehat{\mathbb{V}}}(x,y)={(x-1)-y\over\sqrt{2}},\quad\mbox{for}~{}y\approx
x-1.$

We also observe directly from above that, although ${\bf r}_{\mathbb{V}}={\bf
r}_{\widehat{\mathbb{V}}}=0$ on
$\mathbb{G}_{\mathbb{V}}=\mathbb{G}_{\widehat{\mathbb{V}}}$, their derivatives
are in general not equal.

Consequently, although both (5.11) and (7.25) characterize the same set
$\mathbb{G}_{\mathbb{V}}=\mathbb{G}_{\widehat{\mathbb{V}}}$, the two equations
are fundamentally different. This is partially explained by the above
observation that ${\bf r}_{\mathbb{V}}$ and ${\bf r}_{\widehat{\mathbb{V}}}$
have different derivatives on $\mathbb{G}_{\mathbb{V}}$. At below we provide
more detailed calculation for the set valued heat equation in Example 2.8
(ii), but with $d=m=1$.

###### Example 7.5

Set $d=m=1$, $b=0$, $\sigma=1$, $f=0$, and the terminal
$\mathbb{G}(x)=[-\psi(x),\psi(x)]$, where $\psi:\mathbb{R}\to(0,\infty)$ is
smooth. Then, similar to Example 2.8 (ii), we see that

$\displaystyle\mathbb{V}(t,x)=[-u(t,x),u(t,x)],\quad\mathbb{V}_{b}(t,x)=\\{-u(t,x),u(t,x)\\},$

where $u$ is the unique classical solution of the heat equation

$\displaystyle\partial_{t}u+{1\over 2}\partial_{xx}u=0,\quad u(T,x)=\psi(x).$
(7.30)

We shall prove in Appendix that $\widehat{\bf r}:={\bf
r}_{\widehat{\mathbb{V}}}$ satisfies the following equation:

$\displaystyle\nabla_{t}\widehat{\bf r}+{1\over
2}\Big{[}\nabla_{xx}\widehat{\bf r}-2\nabla_{xy}\widehat{\bf
r}{\nabla_{x}\widehat{\bf r}\over\nabla_{y}\widehat{\bf
r}}+\nabla_{yy}\widehat{\bf r}\big{|}{\nabla_{x}\widehat{\bf
r}\over\nabla_{y}\widehat{\bf
r}}\big{|}^{2}\Big{]}=0,\quad\mbox{on}~{}\mathbb{G}_{\widehat{\mathbb{V}}}.$
(7.31)

In this scalar case, by Remark 5.1 (i) we see that the set valued HJB equation
(5.11) reduces back to the standard PDE for $\overline{v}(t,x)=u(t,x)$ and
$\underline{v}(t,x)=-u(t,x)$, both of which identify with the heat equation
(7.30). So (5.11) is indeed a natural extension of the HJB equation to the
multivariate case. The equation (7.31), however, is quite different from
(7.30). So in this sense, it is more natural to study (5.11) than to study
(7.25).

Another advantage of (5.11) is that, as we saw in Section 7.1, the normal
vector ${\bf n}_{\mathbb{V}}(t,X^{*}_{t},{\Upsilon}^{*}_{t})$ provides
naturally a moving scalarization for the time inconsistent multivariate
optimization problem. The vector ${\bf n}_{\widehat{\mathbb{V}}}$ (at certain
optimal paths) does not serve for this purpose. In fact, ${\bf
n}_{\widehat{\mathbb{V}}}\in\mathbb{R}^{d+m}$, while a moving scalarization
$\Lambda$ is by nature $m$-dimensional.

Finally, we remark that [19, Theorem 2.1] showed that $\widehat{\mathbb{V}}$
is the unique classical solution of (7.25) under the existence of optimal
controls, in the same spirit of our Theorem 6.1 (ii). We instead proved the
existence and uniqueness under weak conditions in Theorems 5.6 and 6.1 (i).

###  7.4 Comparison with Ararat-Ma-Wu [1]

Mainly motivated by dynamic set valued risk measures for multi-asset or
network-based financial models, [1] studied the following set valued BSDE:

$\displaystyle\mathbb{Y}_{t}=\mathbb{E}\Big{[}\mathbb{G}(B_{T})+\int_{t}^{T}\mathbb{F}(s,B_{s},\mathbb{Y}_{s})ds\big{|}{\cal
F}_{t}\Big{]}.$ (7.32)

Here, denoting by ${\cal D}^{m}_{cc}$ the space of convex compact sets
$\mathbb{D}\in{\cal D}^{m}_{0}$, the terminal
$\mathbb{G}:\mathbb{R}^{d}\to{\cal D}^{m}_{cc}$, and the driver
$\mathbb{F}:[0,T]\times\mathbb{R}^{d}\times{\cal D}^{m}_{cc}\to{\cal
D}^{m}_{cc}$ (abusing the notation $\mathbb{F}$ here). We note that [1]
actually allows $\mathbb{G}$ and $\mathbb{F}$ to depend on the paths of $B$.
By relying on the sophisticated set valued stochastic analysis, especially the
Hukuhara difference, [1] established the wellposedness of the above set valued
BSDE. The general case that $\mathbb{F}$ depends on $\mathbb{Z}$, and the
martingale representation with the term $\mathbb{Z}_{t}dB_{t}$ seems to be a
quite remote goal.

Formally, the set valued BSDE (7.32) is associated with our set valued HJB
equation (5.8) in the case $x_{0}=0$, $b=0$, $\sigma=0$, $f=f(t,x,y,a)$. Then
$X_{t}=B_{t}$, and we may naturally define

$\displaystyle\mathbb{Y}_{t}:=\mathbb{V}(t,B_{t}),\quad\mathbb{F}(t,x,\mathbb{D}):=\big{\\{}f(t,x,y,a):y\in\mathbb{D},a\in
A\big{\\}}.$ (7.33)

In the linear case: $f=f(t,x,a)$ and thus
$\mathbb{F}(t,x)=\big{\\{}f(t,x,a):a\in A\big{\\}}$, the random set valued
process $\mathbb{Y}_{t}:=\mathbb{V}(t,B_{t})$ indeed satisfies (7.32) in the
sense of [1].

However, when $f$ depends on $y$, the $\mathbb{Y},\mathbb{F}$ in (7.33) do not
satisfy (7.32). That is, (7.32) is not the stochastic counterpart of (5.8).
The reason is the same as in Remark 2.3 (ii). In (4.3), the $Y$ in the left
side and that in the right side of the BSDE are required to be the same
process. In (7.32), however, one allows to consider different selectors for
the $\mathbb{Y}$ in the left side and that in the right side of the equation.

##  8 Appendix

Proof of Proposition 2.4. Again we denote $r,{\bf n}$ for notational
simplicity. We prove it only for $x>x_{0}$. Fix $x_{1}>x_{0}$. Without loss of
generality, we assume $\theta$ is absolutely continuous in $x\in[x_{0},x_{1}]$
with appropriate derivative function $\theta^{\prime}$, as otherwise the
length of $\theta$ would be $\infty$. Thus we have

$\displaystyle\theta(x)=y_{0}-\int_{x_{0}}^{x}\theta^{\prime}(\tilde{x})d\tilde{x},$

Note that ${\bf r}(x,\theta(x))=0$. Then, for Lebesgue-a.e. $x$,

$\displaystyle 0={d\over dx}{\bf r}(x,\theta(x))=\nabla_{x}{\bf
r}(x,\theta(x))-\nabla_{y}{\bf
r}(x,\theta(x))\cdot\theta^{\prime}(x)=\nabla_{x}{\bf r}(x,\theta(x))-{\bf
n}(x,\theta(x))\cdot\theta^{\prime}(x),$

and thus

$\displaystyle\zeta(x):=\theta^{\prime}(x)-\nabla_{x}{\bf r}{\bf
n}(x,\theta(x))\in\mathbb{T}_{\mathbb{V}}(x,\theta(x)).$

Therefore,

$\displaystyle\displaystyle
L_{\theta}(x_{0},x_{1})=\int_{x_{0}}^{x_{1}}\sqrt{1+|\theta^{\prime}(x)|^{2}}~{}dx=\int_{x_{0}}^{x_{1}}\sqrt{1+\big{|}\nabla_{x}{\bf
r}{\bf n}(x,\theta(x))+\zeta(x)\big{|}^{2}}~{}dx$
$\displaystyle=\int_{x_{0}}^{x_{1}}\sqrt{1+\big{|}\nabla_{x}{\bf
r}(x,\theta(x))\big{|}^{2}+|\zeta(x)|^{2}}~{}dx\geq\int_{x_{0}}^{x_{1}}\sqrt{1+\big{|}\nabla_{x}{\bf
r}(x,\theta(x))\big{|}^{2}}~{}dx$

This implies that

$\displaystyle\displaystyle\mathop{\overline{\rm lim}}_{x_{1}\downarrow
x_{0}}{1\over
x_{1}-x_{0}}\Big{[}L_{\Upsilon}(x_{0},x_{1})-L_{\theta}(x_{0},x_{1})\Big{]}$
$\displaystyle\displaystyle\leq\mathop{\overline{\rm lim}}_{x_{1}\downarrow
x_{0}}{1\over
x_{1}-x_{0}}\Big{[}\int_{x_{0}}^{x_{1}}\sqrt{1+\big{|}\nabla_{x}{\bf
r}(x,{\Upsilon}(x))\big{|}^{2}}~{}dx-\int_{x_{0}}^{x_{1}}\sqrt{1+\big{|}\nabla_{x}{\bf
r}(x,\theta(x))\big{|}^{2}}~{}dx\Big{]}$
$\displaystyle\displaystyle\leq\mathop{\overline{\rm lim}}_{x_{1}\downarrow
x_{0}}{1\over
x_{1}-x_{0}}\Big{[}\int_{x_{0}}^{x_{1}}\big{|}\sqrt{1+\big{|}\nabla_{x}{\bf
r}(x,{\Upsilon}(x))\big{|}^{2}}-\sqrt{1+\big{|}\nabla_{x}{\bf
r}(x_{0},y_{0})\big{|}^{2}}\big{|}~{}dx$
$\displaystyle\displaystyle\qquad\qquad\qquad+\int_{x_{0}}^{x_{1}}\big{|}\sqrt{1+\big{|}\nabla_{x}{\bf
r}(x,\theta(x))\big{|}^{2}}-\sqrt{1+\big{|}\nabla_{x}{\bf
r}(x_{0},y_{0})\big{|}^{2}}\big{|}~{}dx\Big{]}=0.$

Proof of Lemma 2.6. Recall (2.6) and consider the natural extension
$\widehat{\bf n}=\nabla_{y}{\bf r}$. By (2.32), (2.27), and (2.24) we have,
for $i,j=1,\cdots,d$, and $(t,x,y)\in\mathbb{G}_{\mathbb{V}}$,

$\displaystyle\displaystyle\partial_{x_{i}x_{j}}\mathbb{V}(t,x,y)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!-\partial_{x_{i}}\big{(}\nabla_{x_{j}}{\bf r}{\bf
n}\big{)}(t,x,y)=-\Big{[}\partial_{x_{i}}(\nabla_{x_{j}}{\bf r}){\bf
n}+\nabla_{x_{j}}{\bf r}\partial_{x_{i}}{\bf n}\Big{]}(t,x,y)$ (8.1)
$\displaystyle=$ $\displaystyle\\!\\!\\!-\Big{[}\nabla_{x_{i}x_{j}}{\bf
r}+\nabla_{x_{i}}r\nabla_{x_{j}y}{\bf r}\cdot{\bf n}\Big{]}{\bf
n}(t,x,y)-\nabla_{x_{j}}{\bf r}\partial_{x_{i}}{\bf n}(t,x,y).$

Recall (2.6), at $(t,x,y)\in O_{\varepsilon}(\mathbb{G}_{\mathbb{V}})$ we have

$\displaystyle\nabla_{x_{j}y}{\bf r}\cdot\nabla_{y}{\bf r}={1\over
2}\nabla_{x_{j}}\big{(}|\nabla_{y}{\bf r}|^{2}\big{)}=0.$

In particular, $\nabla_{x_{j}y}{\bf r}\cdot{\bf n}(t,x,y)=0$ for
$(t,x,y)\in\mathbb{G}_{\mathbb{V}}$. Plugging this into (8.1) we obtain (2.34)
immediately.

Moreover, again considering the extension $\widehat{\bf
n}^{i}=\nabla_{y_{i}}{\bf r}$, by (2.24) and (2.11) we have

$\displaystyle\partial_{x}n^{i}=\nabla_{xy_{i}}{\bf r}-\nabla_{x}{\bf
r}(\nabla_{y_{i}y}r\cdot{\bf n}),\qquad\partial_{y}{\bf
n}^{i}=\nabla_{y_{i}y}r-(\nabla_{y_{i}y}r\cdot{\bf n}){\bf n}.$

Similarly, by (2.6) we have $\nabla_{y_{i}y}r\cdot{\bf n}=0$, which implies
(2.35) immediately.

Proof of Example 5.3. We first prove (5.14). Denote

$\displaystyle\tilde{f}_{1}(a,y):=a_{1},\quad\tilde{f}_{2}(a,y):=a_{2},\quad\tilde{Y}^{\alpha}_{t}=\int_{t}^{T}\tilde{f}(\alpha_{s},\tilde{Y}^{\alpha}_{s})ds.$

Then one can easily check that

$\displaystyle\left.\begin{array}[]{c}\displaystyle\tilde{V}(t):=\big{\\{}\tilde{Y}^{\alpha}_{t}:\alpha\in{\cal
A}_{t}\big{\\}}=\big{\\{}\tilde{y}\in\mathbb{R}^{2}:|\tilde{y}|\leq
T-t\big{\\}},\\\
\displaystyle\tilde{\mathbb{V}}_{b}(t):=\big{\\{}\tilde{y}\in\mathbb{R}^{2}:|\tilde{y}|=T-t\big{\\}}=\big{\\{}(T-t)\big{(}\cos\theta,\sin\theta\big{)}^{\top}:\theta\in[0,2\pi)\big{\\}}.\end{array}\right.$
(8.4)

Consider a function $\psi:\mathbb{R}^{2}\to\mathbb{R}^{2}$ and set

$\displaystyle
Y^{\alpha}_{t}:=\psi(\tilde{Y}^{\alpha}_{t}),\quad\mbox{where}\quad\psi_{1}(\tilde{y}):=\tilde{y}_{1},\quad\psi_{2}(\tilde{y}):=[1+|\tilde{y}_{1}|^{2}]\tilde{y}_{2}.$

Then we have

$\displaystyle\left.\begin{array}[]{lll}\displaystyle
dY^{\alpha,1}_{t}=d\tilde{Y}^{\alpha,1}_{t}=-\alpha^{1}_{t}dt=-f_{1}(\alpha_{t},Y^{\alpha}_{t})dt;\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle
dY^{\alpha,2}_{t}=2\tilde{Y}^{\alpha,1}_{t}\tilde{Y}^{\alpha,2}_{t}d\tilde{Y}^{\alpha,1}_{t}+[1+|\tilde{Y}^{\alpha,1}_{t}|^{2}]d\tilde{Y}^{\alpha,2}_{t}=-\Big{[}2\tilde{Y}^{\alpha,1}_{t}\tilde{Y}^{\alpha,2}_{t}\alpha^{1}_{t}+[1+|\tilde{Y}^{\alpha,1}_{t}|^{2}]\alpha^{2}_{t}\Big{]}dt\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\qquad\quad=-\Big{[}{2Y^{\alpha,1}_{t}Y^{\alpha,2}_{t}\over
1+|Y^{\alpha,1}_{t}|^{2}}\alpha^{1}_{t}+[1+|Y^{\alpha,1}_{t}|^{2}]\alpha^{2}_{t}\Big{]}dt=-f_{2}(\alpha_{t},Y^{\alpha}_{t})dt.\end{array}\right.$

That is, $Y^{\alpha}$ satisfies (4.3) for given $f$. Therefore,
$\mathbb{V}(t)=\big{\\{}\psi(\tilde{y}):\tilde{y}\in\tilde{\mathbb{V}}(t)\big{\\}}$.
Note further that $\psi_{1}(\tilde{y}_{1})=\tilde{y}_{1}$, and $\psi_{2}$ is
strictly increasing in $\tilde{y}_{2}$. It is clear that

$\displaystyle\mathbb{V}_{b}(t)=\big{\\{}\psi(\tilde{y}):\tilde{y}\in\tilde{\mathbb{V}}_{b}(t)\big{\\}}.$

Plug (8.4) into it, we obtain (5.14) immediately.

We next analyze the convexity of $\mathbb{V}(t)$. Assume for simplicity that
$t=0$. Note that

$\displaystyle\displaystyle\tilde{\mathbb{V}}_{b}(0)=\Big{\\{}\big{(}y_{1},\sqrt{T^{2}-|y_{1}|^{2}}\big{)},~{}\big{(}y_{1},-\sqrt{T^{2}-|y_{1}|^{2}}\big{)}:|y_{1}|\leq
T\Big{\\}};$
$\displaystyle\displaystyle\mathbb{V}_{b}(0)=\Big{\\{}\big{(}y_{1},\varphi(y_{1})\big{)},\big{(}y_{1},-\varphi(y_{1})\big{)}:|y_{1}|\leq
T\Big{\\}},~{}\mbox{where}~{}\varphi(y_{1}):=[1+|y_{1}|^{2}]\sqrt{T^{2}-|y_{1}|^{2}}.$

One may compute straightforwardly that:

$\displaystyle\varphi"(y_{1})={6|y_{1}|^{4}-9T^{2}|y_{1}|^{2}+2T^{4}-T^{2}\over(T^{2}-|y_{1}|^{2})^{3\over
2}},\quad|y_{1}|<T.$

Note that

$\displaystyle\sup_{|y_{1}|<T}\big{[}6|y_{1}|^{4}-9T^{2}|y_{1}|^{2}\big{]}=0.$

So when $T\leq{1\over\sqrt{2}}$ and thus $2T^{4}-T^{2}\leq 0$, we have
$\varphi"(y_{1})\leq 0$ for $|y_{1}|<T$, and in this case $\mathbb{V}(0)$ is
indeed convex. However, when $T>{1\over\sqrt{2}}$, we find that
$\varphi"(y_{1})<0$ for $|y_{1}|\approx T$, but
$\varphi"(0)={2T^{4}-T^{2}\over T^{3}}>0$, then $\mathbb{V}(0)$ is nonconvex.

Proof of Lemma 5.5. Denote

$\displaystyle\tau_{\delta}:=\inf\\{t\geq 0:|{\bf
r}(t,X^{\alpha}_{t},Y^{\alpha}_{t})|\geq\delta\\}\wedge T,$

and consider the linear BSDE with solution pair $(\kappa,\beta)$:

$\displaystyle\kappa_{t}={\bf 1}_{\\{\tau_{\delta}\leq
T_{0}\\}}-\int_{t}^{\tau_{\delta}}\beta_{s}dB_{s},\quad 0\leq t\leq\tau,$
(8.6)

where $\kappa\in\mathbb{R}$, $\beta\in\mathbb{R}^{1\times d}$. It is clear
that $|\kappa|\leq 1$ and $\int_{0}^{\cdot}\beta_{s}dB_{s}$ is an BMO
martingale. Thus,

$\displaystyle\mathbb{E}\Big{[}\exp\big{(}c_{0}\int_{0}^{T}|\beta_{t}|^{2}dt\big{)}\Big{]}\leq
C_{0}<\infty,\quad\mbox{for some $c_{0},C_{0}>0$}.$ (8.7)

For $n\geq 1$, we truncate $\beta$ by $n$ and denote it as $\beta^{n}$. Define

$\displaystyle\kappa^{n}_{t}:=\kappa_{0}+\int_{0}^{t}\beta^{n}_{s}dB_{s}.$

Then it is obvious that, for any $p\geq 1$,

$\displaystyle\mathbb{E}\big{[}\sup_{0\leq
t\leq\tau_{\delta}}|\kappa^{n}_{t}|^{p}\big{]}\leq
C_{p}<\infty;\quad\mbox{and}\quad
c^{n}_{p}:=\Big{(}\mathbb{E}\big{[}\sup_{0\leq t\leq
T_{0}}|\kappa^{n}_{t}-\kappa_{t}|^{p}\big{]}\Big{)}^{1\over p}\to
0,~{}\mbox{as}~{}n\to\infty.$ (8.8)

Introduce two random fields $\nu^{n}(t,\omega,y)$ and $\rho^{n}(t,\omega,y)$:

$\displaystyle\displaystyle\nu^{n}_{ij}(t,y)$ $\displaystyle:=$
$\displaystyle-\sum_{k=1}^{m}\int_{0}^{1}\nabla_{z_{kj}}f^{i}(t,X^{\alpha}_{t},Y^{\alpha}_{t},Z^{\alpha}_{t}+\theta
y\beta^{n}_{t},\alpha_{t})y_{k}d\theta;$
$\displaystyle\displaystyle\rho^{n}_{i}(t,y)$ $\displaystyle:=$
$\displaystyle-\int_{0}^{1}\Big{[}\nabla_{y}f^{i}\cdot y+\hbox{\rm
tr$\,$}((\partial_{z}f^{i})^{\top}\nu^{n}(t,y))\Big{]}$
$\displaystyle\displaystyle\big{(}t,X^{\alpha}_{t},Y^{\alpha}_{t}+\theta\kappa^{n}_{t}y,Z^{\alpha}_{t}+y\beta^{n}_{t}+\theta\kappa^{n}_{t}\nu^{n}(t,y),\alpha_{t}\big{)}d\theta;$

where $\nu^{n}=[\nu^{n}_{ij}]_{1\leq i\leq m,1\leq j\leq
d}\in\mathbb{R}^{m\times d}$, $\rho^{n}=[\rho^{n}_{i}]_{1\leq i\leq
m}\in\mathbb{R}^{m}$. One can easily verify that

$\displaystyle\left.\begin{array}[]{c}\displaystyle\nu^{n}(t,y)\beta^{n}_{t}:=f(t,X^{\alpha}_{t},Y^{\alpha}_{t},Z^{\alpha}_{t},\alpha_{t})-f(t,X^{\alpha}_{t},Y^{\alpha}_{t},Z^{\alpha}_{t}+y\beta^{n}_{t},\alpha_{t});\\\
\displaystyle\kappa^{n}_{t}\rho^{n}(t,y):=f(t,X^{\alpha}_{t},Y^{\alpha}_{t},Z^{\alpha}_{t}+y\beta^{n}_{t},\alpha_{t})-f(t,X^{\alpha}_{t},Y^{\alpha}_{t}+\kappa^{n}_{t}y,Z^{\alpha}_{t}+y\beta^{n}_{t}+\kappa^{n}_{t}\nu(t,y),\alpha_{t}).\end{array}\right.$

Moreover, by Assumption 4.4 (ii) we have

$\displaystyle\left.\begin{array}[]{c}\displaystyle|\nu^{n}(t,y)|\leq
C|y|,\quad|\rho^{n}(t,y)|\leq C|y|;\\\
\displaystyle|\nu^{n}(t,y)-\nu^{n}(t,\tilde{y})|\leq
C_{n}\big{[}1+|y|\big{]}|y-\tilde{y}|,\\\
\displaystyle|\rho^{n}(t,y)-\rho^{n}(t,\tilde{y})|\leq
C_{n}\big{[}1+|y|+|\kappa^{n}_{t}|+|\kappa^{n}_{t}||y|^{2}\big{]}|y-\tilde{y}|.\end{array}\right.$
(8.13)

Next, consider the following SDE:

$\displaystyle\eta^{n}_{t}:=\lambda{\bf
n}_{\mathbb{V}}(0,x_{0},\pi(0,x_{0},Y_{0}^{\alpha}))+\int_{0}^{t}\rho^{n}(s,\eta^{n}_{s})ds+\int_{0}^{t}\nu^{n}(s,\eta^{n}_{s})dB_{s},$
(8.14)

where $\lambda>0$ is a small number which will be determined later. By the
standard stopping arguments for stochastic Lipschitz continuous coefficients,
and by the uniform linear growth in the first line of (8.13), the above SDE is
wellposed, and for any $p\geq 1$,

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq t\leq
T_{0}}|\eta^{n}_{t}|^{p}\Big{]}\leq C_{p}|\lambda|^{p},$ (8.15)

where $C_{p}$ does not depend on $n$.

Denote

$\displaystyle\tilde{Y}^{n}_{t}:=Y_{t}^{\alpha}+\kappa^{n}_{t}\eta^{n}_{t},\quad\tilde{Z}^{n}_{t}:=Z_{t}+\kappa^{n}_{t}\nu^{n}_{t}+\eta^{n}_{t}\beta^{n}_{t}.$

Then by the standard Itô formula we have

$\displaystyle\tilde{Y}^{n}_{t}=Y_{\tau_{\delta}}^{\alpha}+\kappa^{n}_{\tau_{\delta}}\eta^{n}_{\tau_{\delta}}+\int_{t}^{\tau_{\delta}}f(s,X_{s}^{\alpha},\tilde{Y}^{n}_{s},\tilde{Z}^{n}_{s},\alpha_{s})ds-\int_{t}^{\tau_{\delta}}\tilde{Z}^{n}_{s}dB_{s}.$
(8.16)

Moreover, introduce the BSDE

$\displaystyle\hat{Y}^{n}_{t}=Y^{\alpha}_{\tau_{\delta}}+\kappa^{n}_{\tau_{\delta}}\eta^{n}_{\tau_{\delta}}{\bf
1}_{\left\\{{|\kappa^{n}_{\tau_{\delta}}|\leq
2,|\eta^{n}_{\tau_{\delta}}|<{\delta\over
2}}\right\\}}+\int_{t}^{\tau_{\delta}}f(s,X_{s}^{\alpha},\hat{Y}^{n}_{s},\hat{Z}^{n}_{s},\alpha_{s})ds-\int_{t}^{\tau_{\delta}}\hat{Z}^{n}_{s}dB_{s}.$

By (8.8), (8.15), and noting that $|\kappa_{\tau_{\delta}}|\leq 1$, it follows
from standard BSDE estimates that

$\displaystyle|\hat{Y}^{n}_{0}-\tilde{Y}^{n}_{0}|^{2}$ $\displaystyle\leq$
$\displaystyle
C\mathbb{E}\Big{[}\big{|}\kappa^{n}_{\tau_{\delta}}\eta^{n}_{\tau_{\delta}}\big{|}^{2}\big{[}{\bf
1}_{\left\\{{|\kappa^{n}_{\tau_{\delta}}-\kappa_{\tau_{\delta}}|>1}\right\\}}+{\bf
1}_{\left\\{{|\eta^{n}_{\tau_{\delta}}|\geq{\delta\over
2}}\right\\}}\big{]}\Big{]}$ $\displaystyle\leq$ $\displaystyle
C\mathbb{E}\Big{[}\big{|}\kappa^{n}_{\tau_{\delta}}\eta^{n}_{\tau_{\delta}}\big{|}^{2}|\kappa^{n}_{\tau_{\delta}}-\kappa_{\tau_{\delta}}|+{1\over\delta^{2}}|\kappa^{n}_{\tau_{\delta}}|^{2}|\eta^{n}_{\tau_{\delta}}|^{4}\Big{]}\leq
Cc^{n}_{2}+C{|\lambda|^{4}\over\delta^{2}}.$

Thus

$\displaystyle|\hat{Y}^{n}_{0}-\tilde{Y}^{n}_{0}|\leq
C\big{[}\sqrt{c^{n}_{2}}+{|\lambda|^{2}\over\delta}\big{]}.$ (8.17)

Note that

$\displaystyle\hat{Y}^{n}_{\tau_{\delta}}=Y^{\alpha}_{\tau_{\delta}}+\kappa^{n}_{\tau_{\delta}}\eta^{n}_{\tau_{\delta}}{\bf
1}_{\left\\{{|\kappa^{n}_{\tau_{\delta}}|\leq
2,|\eta^{n}_{\tau_{\delta}}|<{\delta\over 2}}\right\\}}.$

On $\\{{|\kappa^{n}_{\tau_{\delta}}|\leq
2,|\eta^{n}_{\tau_{\delta}}|<{\delta\over 2}}\\}^{c}$, we have
$\hat{Y}^{n}_{\tau_{\delta}}=Y^{\alpha}_{\tau_{\delta}}\in\mathbb{V}(\tau_{\delta},X^{\alpha}_{\tau_{\delta}})$.
On $\\{{|\kappa^{n}_{\tau_{\delta}}|\leq
2,|\eta^{n}_{\tau_{\delta}}|<{\delta\over 2}}\\}$, noting again that
$Y^{\alpha}_{\tau_{\delta}}\in\mathbb{V}(\tau_{\delta},X^{\alpha}_{\tau_{\delta}})$,
we have ${\bf
r}_{\mathbb{V}}(\tau_{\delta},X^{\alpha}_{\tau_{\delta}},Y^{\alpha}_{\tau_{\delta}})=-\delta$
and $|\kappa^{n}_{\tau_{\delta}}\eta^{n}_{\tau_{\delta}}|\leq\delta$, then
$\hat{Y}^{n}_{\tau_{\delta}}=Y^{\alpha}_{\tau_{\delta}}+\kappa^{n}_{\tau_{\delta}}\eta^{n}_{\tau_{\delta}}\in\mathbb{V}(\tau_{\delta},X^{\alpha}_{\tau_{\delta}})$.
So in both cases
$\hat{Y}^{n}_{\tau_{\delta}}\in\mathbb{V}(\tau_{\delta},X^{\alpha}_{\tau_{\delta}})$.
Then by DPP (4.17) we have $\hat{Y}^{n}_{0}\in\mathbb{V}(0,x_{0})$. Thus, by
(8.17),

$\displaystyle{\bf
r}_{\mathbb{V}}(0,x_{0},\tilde{Y}^{n}_{0})\leq|\tilde{Y}^{n}_{0}-\hat{Y}^{n}_{0}|\leq
C\big{[}\sqrt{c^{n}_{2}}+{|\lambda|^{2}\over\delta}\big{]}.$

On the other hand, note that
$\tilde{Y}^{n}_{0}=Y^{\alpha}_{0}+\kappa_{0}\lambda{\bf
n}_{\mathbb{V}}(0,x_{0},\pi(0,x_{0},Y_{0}^{\alpha}))$, for $\kappa_{0}\lambda$
small we have

$\displaystyle{\bf r}_{\mathbb{V}}(0,x_{0},\tilde{Y}^{n}_{0})={\bf
r}_{\mathbb{V}}(0,x_{0},Y^{\alpha}_{0})+\kappa_{0}\lambda.$

Thus

$\displaystyle\kappa_{0}\lambda={\bf
r}_{\mathbb{V}}(0,x_{0},\tilde{Y}^{n}_{0})-{\bf
r}_{\mathbb{V}}(0,x_{0},Y^{\alpha}_{0})\leq
C\big{[}\sqrt{c^{n}_{2}}+{|\lambda|^{2}\over\delta}\big{]}+\varepsilon.$

Send $n\to\infty$ and set $\lambda:=\sqrt{\varepsilon\delta}$, we obtain
(5.16):

$\displaystyle\mathbb{P}(\tau_{\delta}\leq T_{0})=\kappa_{0}\leq
C{\lambda\over\delta}+{\varepsilon\over\lambda}=C\sqrt{\varepsilon\over\delta}.$

Finally, if $Y^{\alpha}_{0}\in\mathbb{V}_{b}(0,x_{0})$, then $\varepsilon=0$.
We see that $\mathbb{P}(\tau_{\delta}\leq T_{0})=0$ for all $\delta>0$ and all
$T_{0}<T$. This implies immediately that
$Y^{\alpha}_{t}\in\mathbb{V}_{b}(t,X^{\alpha}_{t})$, $0\leq t<T$, a.s.
Moreover, note that $Y^{\alpha}_{T}=g(X^{\alpha}_{T})$ and
$\mathbb{V}(T,x)=\\{g(x)\\}$, we have
$Y^{\alpha}_{T}\in\mathbb{V}_{b}(T,X^{\alpha}_{T})$ as well.

Proof of Example 6.2. As usual we drop the subscript V in ${\bf r}$ and ${\bf
n}$.

(i) We first show that $\mathbb{V}\in C^{1,2}_{0}([0,T)\times\mathbb{R};{\cal
D}^{2}_{2})$. Fix $\delta>0$ and denote $T_{\delta}:=T-\delta$. By Example 2.8
we have, with $u(t,x)=T-t\geq\delta$ there for $t\in[0,T_{\delta}]$,

$\displaystyle{\bf r}(t,x,y)=|y-w(t,x)|-(T-t).$

Then it is clear that $\mathbb{V}\in
C^{1,2}([0,T_{\delta}]\times\mathbb{R};{\cal D}^{2}_{2})$. By (2.44), for
$|y-w(t,x)|=T-t$, we have

$\displaystyle\left.\begin{array}[]{c}\displaystyle{\bf n}={y-w\over
T-t};\quad\partial_{t}\mathbb{V}=\big{[}\nabla_{t}w\cdot{\bf n}-1\big{]}{\bf
n};\quad\partial_{x}\mathbb{V}=\big{[}\nabla_{x}w\cdot{\bf n}\big{]}{\bf
n};\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle\partial_{x}{\bf
n}={1\over T-t}\big{[}-\nabla_{x}w+[{\bf n}\cdot\nabla_{x}w]{\bf
n}\big{]},\quad\partial_{y}{\bf n}={1\over T-t}\big{[}I_{2\times 2}-{\bf
n}{\bf n}^{\top}\big{]};\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\partial_{xx}\mathbb{V}=-{1\over
T-t}\Big{[}\big{[}|\nabla_{x}w|^{2}-\nabla_{xx}w\cdot{\bf
n}+|\nabla_{x}w\cdot{\bf n}|^{2}\big{]}{\bf n}+[\nabla_{x}w\cdot{\bf
n}]\big{[}\nabla_{x}w-(\nabla_{x}w\cdot{\bf n}){\bf
n}\big{]}\Big{]}.\end{array}\right.$

In particular, $c_{T_{0}}={1\over T}$ in (5.15), and thus $\mathbb{V}\in
C^{1,2}_{0}([0,T)\times\mathbb{R};{\cal D}^{2}_{2})$.

(ii) We next verify the conditions in Theorem 6.1 (ii). For any $a\in A$ and
$\zeta\in\mathbb{T}_{\mathbb{V}}(t,x,y)$, by (5.5) we have: at
$(t,x,y)\in\mathbb{G}_{\mathbb{V}}$,

$\displaystyle\displaystyle
h^{0}_{\mathbb{V}}(t,x,y,\partial_{x}\mathbb{V},\partial_{xx}\mathbb{V},a,\zeta)=\frac{1}{2}\partial_{xx}\mathbb{V}-\Big{[}\zeta\cdot\partial_{x}{\bf
n}+\frac{1}{2}\zeta^{\top}\partial_{y}{\bf n}\zeta\Big{]}{\bf n}$
$\displaystyle\displaystyle=-{1\over
2(T-t)}\Big{[}\big{[}|\nabla_{x}w|^{2}-\nabla_{xx}w\cdot{\bf
n}+|\nabla_{x}w\cdot{\bf n}|^{2}\big{]}{\bf n}+[\nabla_{x}w\cdot{\bf
n}]\big{[}\nabla_{x}w-(\nabla_{x}w\cdot{\bf n}){\bf n}\big{]}\Big{]}$
$\displaystyle\displaystyle\quad-{1\over
T-t}\Big{[}-\zeta\cdot\nabla_{x}w+{1\over 2}|\zeta|^{2}\Big{]}{\bf n}.$

Thus

$\displaystyle\displaystyle{\bf n}\cdot
h_{\mathbb{V}}(t,x,y,\partial_{x}\mathbb{V},\partial_{xx}\mathbb{V},a,\zeta)={\bf
n}\cdot
h^{0}_{\mathbb{V}}(t,x,y,\partial_{x}\mathbb{V},\partial_{xx}\mathbb{V},a,\zeta)+{\bf
n}\cdot[f^{0}(t,x)+a]$ $\displaystyle\displaystyle=-{1\over
2(T-t)}\Big{[}|\nabla_{x}w|^{2}-\nabla_{xx}w\cdot{\bf n}+|\nabla_{x}w\cdot{\bf
n}|^{2}-2\zeta\cdot\nabla_{x}w+|\zeta|^{2}\Big{]}+{\bf
n}\cdot\big{[}f^{0}+a\big{]}$ $\displaystyle\displaystyle=-{1\over
2(T-t)}\Big{[}|\zeta-\nabla_{x}w|^{2}-\nabla_{xx}w\cdot{\bf
n}+|\nabla_{x}w\cdot{\bf n}|^{2}\Big{]}+{\bf n}\cdot\big{[}f^{0}+a\big{]}.$

Recall $|a|\leq 1$, then clearly the optimal arguments are:

$\displaystyle a^{*}=I^{\mathbb{V}}_{1}(t,x,y):={\bf
n}(t,x,y),\quad\zeta^{*}=I^{\mathbb{V}}_{2}(t,x,y):=\nabla_{x}w-[\nabla_{x}w\cdot{\bf
n}]{\bf n}.$

Together with (6.4), this implies further that

$\displaystyle\tilde{I}^{\mathbb{V}}_{3}:=-\Big{[}\partial_{t}\mathbb{V}+h^{0}_{\mathbb{V}}(\cdot,\partial_{x}\mathbb{V},\partial_{xx}\mathbb{V},I^{\mathbb{V}}_{1},I^{\mathbb{V}}_{2})+f^{0}+I^{\mathbb{V}}_{1}\Big{]}$
$\displaystyle=-\big{[}\nabla_{t}w\cdot{\bf n}-1\big{]}{\bf n}+{1\over
T-t}\big{[}-\zeta\cdot\nabla_{x}w+{1\over 2}|\zeta|^{2}\big{]}{\bf
n}-[f^{0}+{\bf n}]$ $\displaystyle\quad+{1\over
2(T-t)}\Big{[}\big{[}|\nabla_{x}w|^{2}-\nabla_{xx}w\cdot{\bf
n}+|\nabla_{x}w\cdot{\bf n}|^{2}\big{]}{\bf n}+[\nabla_{x}w\cdot{\bf
n}]\big{[}\nabla_{x}w-(\nabla_{x}w\cdot{\bf n}){\bf n}\big{]}\Big{]};$
$\displaystyle I^{\mathbb{V}}_{3}:=-f^{0}+[{\bf n}\cdot f^{0}]{\bf n}+{1\over
2(T-t)}[\nabla_{x}w\cdot{\bf n}]\big{[}\nabla_{x}w-(\nabla_{x}w\cdot{\bf
n}){\bf n}\big{]}.$

Plug $I^{\mathbb{V}}_{1},I^{\mathbb{V}}_{2},I^{\mathbb{V}}_{3}$ into (6.8),
clearly the resulted SDE is wellposed.

Proof of Example 7.5. We now compute the equation (7.25) in this case. First,

$\displaystyle{\cal N}(t,x,y,\nabla_{x}\eta,\nabla_{y}\eta)$ $\displaystyle=$
$\displaystyle\big{\\{}(a,z):[1,z]^{\top}[\nabla_{x}\eta+z\nabla_{y}\eta]=0\big{\\}}$
$\displaystyle=$
$\displaystyle\big{\\{}(a,z):z=-{\nabla_{x}\eta\over\nabla_{y}\eta}\big{\\}}=\big{\\{}\big{(}a,-{\nabla_{x}\widehat{\bf
r}(t,x,y)\over\nabla_{y}\widehat{\bf r}(t,x,y)}\big{)}\big{\\}}.$

Then, recalling $\eta={1\over 2}|\widehat{\bf r}|^{2}$,

$\displaystyle F$ $\displaystyle=$ $\displaystyle{1\over
2}\Big{[}\nabla_{xx}\eta-2\nabla_{xy}\eta{\nabla_{x}\widehat{\bf
r}\over\nabla_{y}\widehat{\bf
r}}+\nabla_{yy}\eta\big{|}{\nabla_{x}\widehat{\bf
r}\over\nabla_{y}\widehat{\bf r}}\big{|}^{2}\Big{]}(t,x,y)$ $\displaystyle=$
$\displaystyle{1\over 2}\Big{[}\widehat{\bf r}\nabla_{xx}\widehat{\bf
r}+|\nabla_{x}\widehat{\bf r}|^{2}-2[\widehat{\bf r}\nabla_{xy}\widehat{\bf
r}+\nabla_{x}\widehat{\bf r}\nabla_{y}\widehat{\bf r}]{\nabla_{x}\widehat{\bf
r}\over\nabla_{y}\widehat{\bf r}}+[\widehat{\bf r}\nabla_{yy}\widehat{\bf
r}+|\nabla_{y}\widehat{\bf r}|^{2}]\big{|}{\nabla_{x}\widehat{\bf
r}\over\nabla_{y}\widehat{\bf r}}\big{|}^{2}\Big{]}(t,x,y)$ $\displaystyle=$
$\displaystyle{1\over 2}\widehat{\bf r}\Big{[}\nabla_{xx}\widehat{\bf
r}-2\nabla_{xy}\widehat{\bf r}{\nabla_{x}\widehat{\bf
r}\over\nabla_{y}\widehat{\bf r}}+\nabla_{yy}\widehat{\bf
r}\big{|}{\nabla_{x}\widehat{\bf r}\over\nabla_{y}\widehat{\bf
r}}\big{|}^{2}\Big{]}(t,x,y).$

Note further that $\widehat{\bf r}=0$ on $\mathbb{G}_{\mathbb{V}}$. Then, for
$(t,x,y)\in\mathbb{G}_{\mathbb{V}}$, we have

$\displaystyle\nabla_{x}F$ $\displaystyle=$ $\displaystyle{1\over
2}\nabla_{x}\widehat{\bf r}\Big{[}\nabla_{xx}\widehat{\bf
r}-2\nabla_{xy}\widehat{\bf r}{\nabla_{x}\widehat{\bf
r}\over\nabla_{y}\widehat{\bf r}}+\nabla_{yy}\widehat{\bf
r}\big{|}{\nabla_{x}\widehat{\bf r}\over\nabla_{y}\widehat{\bf
r}}\big{|}^{2}\Big{]}(t,x,y);$ $\displaystyle\nabla_{y}F$ $\displaystyle=$
$\displaystyle{1\over 2}\nabla_{y}\widehat{\bf
r}\Big{[}\nabla_{xx}\widehat{\bf r}-2\nabla_{xy}\widehat{\bf
r}{\nabla_{x}\widehat{\bf r}\over\nabla_{y}\widehat{\bf
r}}+\nabla_{yy}\widehat{\bf r}\big{|}{\nabla_{x}\widehat{\bf
r}\over\nabla_{y}\widehat{\bf r}}\big{|}^{2}\Big{]}(t,x,y).$

On the other hand, note that $\nabla_{t}\eta=\widehat{\bf
r}\nabla_{t}\widehat{\bf r}$. Then, again at
$(t,x,y)\in\mathbb{G}_{\mathbb{V}}$,

$\displaystyle\nabla_{x}\nabla_{t}\eta=\nabla_{x}\widehat{\bf
r}\nabla_{t}\widehat{\bf
r},\quad\nabla_{y}\nabla_{t}\eta=\nabla_{y}\widehat{\bf
r}\nabla_{t}\widehat{\bf r}.$

Plug these into (7.25), we obtain (7.31) immediately.

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# Achievement of 200,000 hours of operation at
KEK 7-GeV electron 4-GeV positron injector linac

K Furukawa1,2 M Akemoto1 D Arakawa1 Y Arakida1 Y Bando2 H Ego1,2 Y Enomoto1,2
T Higo1 H Honma1 N Iida1,2 K Kakihara1 T Kamitani1,2 H Katagiri1 M Kawamura1 S
Matsumoto1,2 T Matsumoto1,2 H Matsushita1 K Mikawa1 T Miura1,2 F Miyahara1,2 H
Nakajima1 T Natsui1,2 Y Ogawa1 S Ohsawa1 Y Okayasu1,2 T Oogoe1 M A Rehman1 I
Satake1 M Satoh1,2 Y Seimiya1,2 T Shidara1 A Shirakawa1 H Someya1 T Suwada1,2
M Tanaka1 D Wang1 Y Yano1 K Yokoyama1,2 M Yoshida1,2 T Yoshimoto1,2 R Zhang1,2
and X Zhou1,2 1 High Energy Accelerator Research Organization (KEK), Tsukuba,
Ibaraki, 305-0801, Japan  2 Graduate University for Advanced Studies
(SOKENDAI), Tsukuba, Ibaraki, 305-0801, Japan<EMAIL_ADDRESS>
###### Abstract

KEK electron positron injector LINAC initiated the injection operation into
Photon Factory (PF) light source in 1982. Since then for 39 years, it has
served for multiple projects, namely, TRISTAN, PF-AR, KEKB, and SuperKEKB. Its
total operation time has accumulated 200 thousand hours on May 7, 2020. We are
extremely proud of the achievement following continuous efforts by our
seniors. The construction of the injector LINAC started in 1978, and it was
commissioned for PF with 2.5 GeV electron in 1982. In parallel, the positron
generator linac was constructed for the TRISTAN collider project. The slow
positron facility was also commissioned in 1992. After the KEKB asymmetric-
energy collider project was commissioned in 1998 with direct energy
injections, the techniques such as two-bunch acceleration and simultaneous
injection were developed. As the soft structure design of the LINAC was too
weak against the great east Japan earthquake, it took three years to recover.
Then the construction and commissioning for the SuperKEKB project went on, and
the simultaneous top-up injection into four storage rings contributes to the
both elementary particle physics and photon science.

##  1 Introduction

In the experimental particle physics research in the 1970s, there was growing
expectation for the world-class Japanese domestic collider after the successes
in the world-level theoretical research. On the other hand, in the field of
synchrotron radiation science, there were demands for the establishment of a
synchrotron radiation research institute with a dedicated accelerator. In
order to realize both of those accelerator projects, a 400-m electron linear
accelerator capable of 2.5 GeV direct energy injection was constructed in
1982.

Figure 1:  Electron positron accelerator projects operated by the injector
LINAC with their injection beam energy and experimental beam energy.

Since then, the injector had been in operation for 38 years in 2020,
supporting the consecutive accelerator projects of Photon Factory (PF),
TRISTAN, KEKB, PF-AR, and SuperKEKB as depicted in Figs. 1, 2, 3 and 5. It has
finally achieved 200,000 hours of operation in May 2020. The brief operation
history would be given.

##  2 2.5 GeV injection for PF/TRISTAN

At the time of the initial construction since 1978, several university-based
accelerator facilities had achieved research successes in Japan. It was
fortunate that a new linear electron accelerator had been built to share
resources for advanced scientific research in both particle physics and
synchrotron radiation science, which were based on the accumulated experiences
at universities. The situation might be similar to recent J-PARC’s focus on
both nuclear and particle physics, as well as neutron and muon science. It
seems that there are not many accelerator projects in the world that have
multiple disciplines from the beginning. Those Japanese accelerators are quite
unique that deliver beams for multiple science fields.

While many researchers of synchrotron radiation research wanted the institute
independent from particle physics, the accelerator science played a role as a
bridge between those two. One hundred sixty of 2-m-long S-band traveling-wave
accelerating structures with $2\pi/3$ mode quasi-constant gradient were
installed, and driven by forty 20-MW klystrons to achieve 2.5 GeV electron and
positron beams [1]. 2.5 GeV direct energy injection was made for PF and 2.5
GeV electron positron beams from the injector were accelerated through the
TRISTAN synchrotron chain for the 32 GeV collision as depicted in Fig. 2.

Figure 2:  Injection for TRISTAN and PF.

When the synchrotron radiation facility, PF, was completed, there must be an
enthusiastic response to the dedicated synchrotron radiation facility that
would promote research in synchrotron radiation science. The TRISTAN project
also began to take the lead in high-energy experimental physics, after
Japanese theorist had contributed to the worldwide development of particle
physics theory. It was realized recently that the specifications of the
electric power facilities built at that time secured the expansion possibility
even for the fields of nuclear physics and ultraviolet synchrotron radiation
science. Such design facilitated the upgrades towards KEKB and SuperKEKB
later.

##  3 8-GeV upgrade for KEKB

Since the construction of TRISTAN, a linear collider was expected as the next
particle physics experiment project. Instead of immediately promoting a large
scale collider, however, the construction of a B Factory, KEKB, was carried
out to experimentally confirm the remaining issues of CP violation in particle
theory proposed by Makoto Kobayashi and Toshihide Maskawa. Since the injection
performance was important to achieve the integral performance as a factory,
there were concerns compared to SLAC/PEP-II, which was planning the B-factory
in parallel.

Figure 3:  Injection for KEKB, PF and PF-AR.

The direct energy injections for the asymmetric energy collider rings were
realized by an energy upgrade from 2.5 GeV to 8 GeV by introducing SLED
microwave pulse compressors and adding 50% more accelerating structures that
extends the length of the injector up to 700 m in a shape of a letter ‘J’ as
in Fig. 3. It was a substantial and difficult construction because the
injector continued the PF injection [2, 3]. It made maximum use of the
opportunities of the longest shutdown of nine months in 1997 during the
construction.

The success of the KEKB operation was largely due to the online operating
environment of the SAD beam simulation software and the EPICS control
framework which were developed during the injector commissioning period [4].

KEKB and PEP-II projects developed the friendly competition, and they compared
the performance of the accelerators in those control rooms everyday [5]. The
performance of the KEK injector was gradually improved during the project
step-by-step [6].

Figure 4:  Progress of the daily operation during the KEKB project, with the
stored beam current in red and collision luminosity in yellow.

The positron beam injection was enhanced by employing dual bunch acceleration.
The separation of two bunches was chosen to be 96 ns because of the frequency
relationship between the injector and the ring [7].

Then later, it also succeeded in continuous injection by switching the
injection every several minutes while the Belle collision detector was in
operation. In 2008, the injection was further improved by the simultaneous
top-up injection in to three storage rings except PF-AR in pulse-to-pulse
modulation switching beams with distinct properties every 20 ms employing
event-based control architecture [8] as in Fig. 4.

##  4 Slow positron and others

The injector facility usually contributes by injecting beams into the
downstream storage rings. However, there are cases where experiments are
performed using direct electrons. In the past at KEK LINAC, there have been
experiments at the beam dump for a search for an axion like particle, beam
irradiation calibration of tile fiber calorimeter detectors for the SSC
project, positron beam channeling experiments, and positron production using
electron channeling radiation. As a result of the positron generation
experiment, a crystal target was actually applied to the positron generation
for KEKB injection in 2006, and the positron generation efficiency was
successfully improved by about 25% [9].

Another ongoing project is the research experiments for material science and
particle physics by slow positrons. Between injections to the TRISTAN project
slow positrons were generated at the end of LINAC. While the experimental
setup had to be moved twice due to the change in the beam operation of the
main LINAC, now a dedicated 50 MeV linac is used for slow positron experiments
sharing the same resources as the main LINAC [10]. It is valuable to produce
scientific results within the LINAC building.

##  5 SuperKEKB and simultaneous top-up injection

After the KEKB project was completed in 2010 the injector, whose girders were
built with the concept of a flexible structure, suffered a serious damage due
to the Great East Japan Earthquake in 2011. While light source injections were
recovered in 3 months, it took three years to start the beam test for
SuperKEKB as the alignment was severely deteriorated. Actually, the alignment
requirement became much tight in order to suppress the wakefield effect in the
accelerating structure. While the light source injections were maintained even
during the upgrade towards SuperKEKB, the longest construction period of five
months was secured in 2017.

Figure 5:  Injection for SuperKEKB as well as PF and PF-AR.

The energy enhancement was the main subject for the previous KEKB injection
upgrade, on the other hand the SuperKEKB project required quality improvements
of the injection beam for the electron positron collision with the novel nano-
beam scheme [11] as in Fig. 5. A beam emittance of about 20 mm.mrad was
required with a charge of 4 nC per bunch. The positron capture section was
reconstructed [12] in order to increase the positron charge employing a flux
concentrator pulsed solenoid, large aperture S-band structures (LASs), long DC
solenoid magnets and a hundred of quadrupole focusing magnets. A damping ring
was newly constructed to damp the positron emittance. For the electron beam, a
new high-current and low-emittance RF gun was developed with an iridium-cerium
photocathode and a quasi-travelling-wave side-coupled cavity (QTWSC) [13].

More than hundred pulsed focusing and corrector magnets were newly installed
[14]. Pulsed low-level RF controls were also developed. Based on those
equipment upgrades [15], the simultaneous top-up injections for four storage
rings of SuperKEKB HER, LER, PF ring and PF-AR were realized in 2019, which
contributed to the considerable improvement of experimental efficiency [16].

The continuous operation of the electron-positron injector has supported the
history of advanced research at KEK by providing beams for various projects as
shown in Fig. 1.

Figure 6:  Yearly operation hours of the injector LINAC with failure rates.

At 8:50 a.m. on May 7, 2020, 200,000 hours of operation was accomplished as
shown in Fig. 6, since the PF injection operation started in 1982. Authors are
very proud to have reached the milestone of 200,000 hours of operation by
inheriting the operation results accumulated by our predecessors. Although no
special event was held due to the effects of COVID-19, a toast was made with
morning coffee via video conference. In the statistics, the partial failure
rate indicates that a certain device has failed but the beam injection
operation was still possible due to redundancy, and the injection stop rate
indicates that the failure was so serious that the beam injection was
impossible. At the beginning of each program, the failure rate increases
before understanding the equipment characteristics, but it tends to stabilize
as countermeasures are taken.

##  6 Conclusion

The KEK electron-positron injector LINAC has been able to achieve 200,000
hours of operation while supporting experiments in multiple fields of photon
science and particle physics.

## References

## References

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