# Optimization-based Constrained Funnel Synthesis for Systems with Lipschitz
Nonlinearities via Numerical Optimal Control

Taewan Kim  Purnanand Elango  Taylor P. Reynolds
Behçet Açıkmeşe  Fellow, IEEE  and Mehran Mesbahi  Fellow, IEEE Manuscript
received 17 March 2023; revised 19 May 2023; accepted 13 June 2023. This work
was supported by Air Force Office of Scientific Research grant
FA9550-20-1-0053. (Corresponding author: Taewan Kim.)Taewan Kim, Purnanand
Elango, Behçet Açıkmeşe, and Mehran Mesbahi are with the Department of
Aeronautics and Astronautics, University of Washington, Seattle, WA 98195, USA
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>mesbahi@uw.edu).Taylor
P. Reynolds is with Amazon Prime Air, Seattle, WA 98117, USA
(e-mail:tayreyno@amazon.com).

###### Abstract

This paper presents a funnel synthesis algorithm for computing controlled
invariant sets and feedback control gains around a given nominal trajectory
for dynamical systems with locally Lipschitz nonlinearities and bounded
disturbances. The resulting funnel synthesis problem involves a differential
linear matrix inequality (DLMI) whose solution satisfies a Lyapunov condition
that implies invariance and attractivity properties. Due to these properties,
the proposed method can balance maximization of initial invariant funnel size,
i.e., size of the funnel entry, and minimization of the size of the attractive
funnel for attenuating the effect of disturbance. To solve the resulting
funnel synthesis problem with the DLMI as constraints, we employ a numerical
optimal control approach that uses a multiple shooting method to convert the
problem into a finite dimensional semidefinite programming problem. This
framework does not require piecewise linear system matrices and funnel
parameters, which is typically assumed in recent related work. We illustrate
the proposed funnel synthesis method with a numerical example.

Lyapunov methods, LMIs, Robust control

##  1 INTRODUCTION

Funnel, also referred to as tube, represents regions of finite-time controlled
invariant state space for closed-loop systems equipped with an associated
feedback control law around a given nominal trajectory [1]. Funnel synthesis
refers to a procedure for computing both the controlled invariant set and the
corresponding feedback control law. Once we compute a library of funnels along
different nominal trajectories, the resulting funnel can be used for different
purposes such as real-time motion planning [2] and feasible trajectory
generation [3].

The studies in funnel synthesis can be separated into two categories depending
on whether they aim to maximize [3, 4, 5] or minimize the size of the funnel
[2, 6]. The funnel computation inherently aims to maximize the size of the
funnel to have a larger controlled invariant set in the state space. On the
other hand, when it comes to systems under uncertainty or disturbances, the
funnel size should be minimized to bound the effect of the uncertainty. For
example, [2] minimize the size of the funnel to prohibit collision with
obstacles instead of imposing obstacle avoidance constraints directly.
However, minimizing the size of the funnel is against the original purpose of
having a large controlled invariant set in the state space. In this work, we
provide a funnel synthesis algorithm that balances maximizing the size of the
funnel and minimizing the effect of the bounded disturbance. To this end, we
exploit invariance and attractivity conditions derived from Lyapunov theory
[7] by solving linear matrix inequalities (LMIs) [8, 9] and imposing state and
input constraints directly on the funnel.

When employing the Lyapunov condition, the resulting optimization problem has
a differential inequality of the Lyapunov function in continuous-time for a
finite-time interval. Since it is intractable to satisfy the inequality for
all time in the given interval, many approaches focus on imposing the
differential inequality at a finite number of node points [2, 4, 5]. When a
quadratic Lyapunov function with a time-varying positive definite (PD) matrix
is employed, the resulting differential inequality ends up with a differential
linear matrix inequality (DLMI). To solve the resulting DLMI, one can assume
that first-order approximations (Jacobians matrices) of the nonlinear dynamics
computed around the nominal trajectory are continuous piecewise linear in
time. By applying the same piecewise linear parametrization to the PD matrix
in the Lyapunov function, one can obtain a finite number of LMIs whose
feasibility is a sufficient condition for the original DLMI [3, 10]. The main
downside of this approach is that the assumption of piecewise linear system
matrices may have large errors, and applying the same parametrization on the
PD matrix can be conservative.

In this paper, we provide a constrained funnel synthesis algorithm for locally
Lipschitz nonlinear systems under bounded disturbance. To this end, we express
the closed-loop system around the given nominal trajectory as a linear time-
varying system having uncertain terms. Then, the DLMI is derived based on the
Lyapunov condition that guarantees the invariance and the attractivity
conditions. With the Lyapunov condition, the continuous-time funnel
optimization problem maximizes the size of the funnel entry and minimizes the
attractive funnel for attenuating the effect of disturbance. Furthermore, the
proposed method can satisfy linear state and control constraints in a way that
the resulting funnel around the given nominal trajectory remains inside the
feasible sets of states and controls. To convert the funnel synthesis problem
into a finite dimensional semidefinite programming (SDP) problem, we employ a
numerical optimal control approach with a multiple shooting method [11].

The contributions of this work are as follows: First, the proposed funnel
synthesis approach for locally Lipschitz nonlinear systems provides a new
optimization framework that can 1) balance maximizing the size of the funnel
entry and minimizing the effect of the disturbance, and 2) guarantee the
satisfaction of linear state and control constraints on the funnel. Second, we
provide a new approach based on multiple shooting in numerical optimal control
for solving the DLMI. This is in contrast to the prior approaches that assume
the piecewise linear approximation on the PD matrix in the Lyapunov function.

The notations $\mathbb{R}$, $\mathbb{R}_{+}$ $\mathbb{R}_{++}$, and
$\mathbb{R}^{n}$ are the field of real, nonnegative, positive numbers, and the
$n$-dimensional Euclidean space, respectively. The set $\mathcal{N}_{q}^{r}$
is a finite set of consecutive nonnegative integers, i.e.,
$\\{q,q+1,\ldots,r\\}$. The symmetric matrix $Q=Q^{\top}(\succeq)\\!\succ 0$
implies $Q$ is PD (PSD) matrix, and
$(\mathbb{S}^{n}_{+})\,\mathbb{S}^{n}_{++}$ denotes the set of all PD (PSD)
matrices whose size is $n\times n$. The symbols $\otimes$ is the Kronecker
product. The notation * denotes the symmetric part of a matrix. The squared
root of a PSD matrix $A$ is defined as $A^{\frac{1}{2}}$ such that
$A=A^{\frac{1}{2}}A^{\frac{1}{2}}$. We omit the time argument $t$ if it is
clear from the context. The operation $\oplus$ is Minkowski sum.

##  2 Constrained Funnel Synthesis

###  2.1 Locally Lipschitz Nonlinear Systems

Consider the following continuous-time dynamics:

$\displaystyle\dot{x}(t)$ $\displaystyle=f(t,x(t),u(t),w(t)),\quad
t\in[t_{0},t_{f}],$ (1)

where $x(t)\in\mathbb{R}^{n_{x}}$ is state and $u(t)\in\mathbb{R}^{n_{u}}$ is
input. The vector-valued function $w(t)\in\mathbb{R}^{n_{w}}$ represents
bounded disturbance such that $\lVert w(\cdot)\rVert_{\infty}\leq 1$ where
$\lVert w(\cdot)\rVert_{\infty}\coloneqq\sup_{t\in[t_{0},t_{f}]}\lVert
w(t)\rVert$, and $t_{0}$ and $t_{f}$ are initial and final time, respectively.
The function
$f:\mathbb{R}_{+}\times\mathbb{R}^{n_{x}}\times\mathbb{R}^{n_{u}}\times\mathbb{R}^{n_{w}}\rightarrow\mathbb{R}^{n_{x}}$
is assumed to be continuously differentiable. Suppose that a nominal
trajectory $(\bar{x}(\cdot),\bar{u}(\cdot),\bar{w}(\cdot))$ is a solution of
the system (1). Particularly, we choose a zero disturbance for the nominal
trajectory, that is $\bar{w}(t)=0$ for all $t\in[t_{0},t_{f}]$. Then, we can
convert (1) into the linear time-varying (LTV) system with the nonlinear
remainder term via linearization around the nominal trajectory, resulting in
the following Lur’e type system [3, 12]:

$\displaystyle\dot{x}(t)$ $\displaystyle=A(t)x(t)+B(t)u(t)+F(t)w(t)+Ep(t),$
$\displaystyle p(t)$ $\displaystyle=\phi(t,q(t)),\quad
q(t)=Cx(t)+Du(t)+Gw(t),$ (2)

where $p(t)\in\mathbb{R}^{n_{p}}$ is a lumped nonlinearity represented by a
nonlinear function $\phi(t)$ and its argument $q(t)\in\mathbb{R}^{n_{q}}$. The
matrices $A(t)$, $B(t)$, and $F(t)$ are first-order approximations of the
nonlinear dynamics (1) around the nominal trajectory. The matrices
$E\in\mathbb{R}^{n_{x}\times n_{p}}$, $C\in\mathbb{R}^{n_{q}\times n_{x}}$,
$D\in\mathbb{R}^{n_{q}\times n_{u}}$, and $G\in\mathbb{R}^{n_{q}\times n_{w}}$
are assumed to be time-invariant 111 The matrices $E,C,D$, and $G$ are
selector matrices with 0s and 1s to organize the nonlinearity of the system.
The simplest case has $E=I$, $q=[x^{\top},u^{\top},w^{\top}]^{\top}$, and
$p=\phi(t,q)=f(t,x,u,w)-Ax-Bu-Fw$.. The more details in choosing these
matrices could be found in [3].

With the state difference $\eta\coloneqq x-\bar{x}$, the difference dynamics
can be derived as

$\displaystyle\dot{\eta}(t)$
$\displaystyle=f(t,x,u,w)-f(t,\bar{x},\bar{u},0),$
$\displaystyle=A(t)\eta(t)+B(t)\xi(t)+F(t)w(t)+E\delta p(t),$
$\displaystyle\delta p(t)$ $\displaystyle=\phi(t,q(t))-\phi(t,\bar{q}(t)),$
$\displaystyle\delta q(t)$ $\displaystyle=C\eta(t)+D\xi(t)+Gw(t),$

where $\xi\coloneqq u-\bar{u}$ and $\delta q\coloneqq q-\bar{q}$ with
$\bar{q}=C\bar{x}+D\bar{u}$. Since continuously differentiable functions are
locally Lipschitz, $f$ and $\phi$ are locally Lipschitz. It follows that for
all $t\in[t_{0},t_{f}]$

$\displaystyle\lVert p-\bar{p}\rVert_{2}$ $\displaystyle\leq\gamma(t)\lVert
q-\bar{q}\rVert_{2},\quad\forall\,q,\bar{q}\in\mathcal{Q},$

where $\gamma(t)\in\mathbb{R}_{+}$ is a Lipschitz constant for each $t$ and
$\mathcal{Q}\subseteq\mathbb{R}^{n_{x}}$ is any compact set. By employing the
linear feedback controller, that is $\xi(t)=K(t)\eta(t)$, the closed-loop
system can be written as

$\displaystyle\dot{\eta}$ $\displaystyle=(A+BK)\eta+Fw+E\delta p,\quad\lVert
w(\cdot)\rVert_{\infty}\leq 1,$ (3a) $\displaystyle\delta q$
$\displaystyle=C\eta+D\xi+Gw,\quad\lVert\delta
p\rVert_{2}\leq\gamma\lVert\delta q\rVert_{2}.$ (3b)

With (3), we can express the nonlinear closed-loop system as the LTV system
having the state and input dependent uncertainty $\delta p$. This could be a
conservative way to handle the nonlinear system, but it allows us to design a
quadratic Lyapunov function with which we can guarantee the invariance and
attractivity conditions of the funnel for the original nonlinear system.

###  2.2 Lyapunov Conditions

With a continuously differentiable positive definite matrix-valued function
$Q:\mathbb{R}_{+}\rightarrow\mathbb{S}_{++}^{n_{x}}$, the Lyapunov function is
defined as

$\displaystyle V(t,\eta)$
$\displaystyle\coloneqq\eta^{\top}(t)Q^{-1}(t)\eta(t).$ (4)

Here we aim to impose the following Lyapunov condition for the closed-loop
system (3):

$\displaystyle\dot{V}(t,\eta)$ $\displaystyle\leq-\alpha V(t,\eta),$ (5a)
$\displaystyle\text{for all }\lVert\delta p(t)\rVert_{2}$
$\displaystyle\leq\gamma(t)\lVert\delta q(t)\rVert_{2},$ (5b)
$\displaystyle\text{and }V(t,\eta)$ $\displaystyle\geq\lVert
w(t)\rVert_{2}^{2},\quad\forall\,t\in[t_{0},t_{f}],$ (5c)

where $\alpha\in\mathbb{R}_{++}$ is a decay rate. With the above Lyapunov
condition, we can establish the following lemma.

######  Lemma 1.

Suppose that the Lyapunov condition (5) holds with a positive definite matrix-
valued continuous function $Q(t)$ , then the time-varying ellipsoid defined as

$\displaystyle\mathcal{E}(t)=\\{\eta\mid\eta^{\top}Q(t)^{-1}\eta\leq 1\\},$
(6)

is invariant for the closed-loop system (3), that is, if $\eta(\cdot)$ is any
solution with $\eta(t_{0})\in\mathcal{E}(t_{0})$, then
$\eta(t)\in\mathcal{E}(t)$ for all $t\in[t_{0},t_{f}]$. Furthermore, the
ellipsoid $\mathcal{E}(t)$ is attractive such that for any solution
$\eta(\cdot)$, the following holds for all $t\in[t_{0},t_{f}]$:

$\displaystyle
V(t,\eta(t))\leq\max\\{e^{-\alpha(t-t_{0})}V(t_{0},\eta(t_{0})),1\\}.$ (7)

The above lemma can be deduced from [9, Lemma B10] and [13, Lemma 1], so here
we skip the proof.

Now we define a invariant state funnel with a pair of $Q$ in (4) and a
continuous scalar-valued function $c:\mathbb{R}_{+}\rightarrow(0,1]$ as

$\displaystyle\mathcal{E}_{c}(t)\coloneqq\left\\{\,\eta\,\middle|\,\eta^{\top}Q(t)^{-1}\eta\leq\frac{1}{c(t)}\right\\},$
(8)

where the function $c(t)$ satisfy the following condition:

$\displaystyle\frac{1}{c(t)}\geq\max\left\\{1,e^{-\alpha(t-t_{0})}\frac{1}{c(t_{0})}\right\\},$
(9)

with $0<c(t_{0})\leq 1$. With the ellipsoid $\mathcal{E}_{c}(t)$ having
$1/c(t)$ as the support value, we show the invariance property of
$\mathcal{E}_{c}(t)$ in the following lemma.

######  Lemma 2.

The ellipsoid $\mathcal{E}_{c}(t)$ defined in (8) with (9) is invariant for
the closed-loop system (3) such that if $\eta(\cdot)$ is any solution with
$\eta(t_{0})\in\mathcal{E}_{c}(t_{0})$, then $\eta(t)\in\mathcal{E}_{c}(t)$
for all $t\in[t_{0},t_{f}]$.

###### Proof.

If the solution $\eta(\cdot)$ satisfies $\eta(t_{0})\in\mathcal{E}(t_{0})$, it
is trivial to prove the invariance of $\mathcal{E}_{c}(t)$ since
$\mathcal{E}(t)$ is invariant and $\mathcal{E}(t)\subseteq\mathcal{E}_{c}(t)$.
Consider the solution $\eta(\cdot)$ such that
$\eta(t_{0})\in\mathcal{E}_{c}(t_{0})\setminus\mathcal{E}(t_{0})$. By the
attractivity condition (7), we have
$V(t,\eta(t))\leq\max\\{e^{-\alpha(t-t_{0})}V(t_{0},\eta(t_{0})),1\\}$. It
follows from $V(t_{0},\eta(t_{0}))\leq 1/c(t_{0})$ and $0<c(t_{0})\leq 1$ that
$V(t,\eta(t))\leq\max\\{e^{-\alpha(t-t_{0})}\frac{1}{c(t_{0})},1\\}\leq
1/c(t)$ for $t\in[t_{0},t_{f}]$. This completes the proof. ∎

Figure 1:  Illustration of the ellipsoids $\mathcal{E}(t)$ and
$\mathcal{E}_{c}(t)$. An example of solution $\eta(t)$ is given as a dashed
red line. Since $\mathcal{E}$ is attractive, any solution $\eta(\cdot)$
starting with
$\eta(t_{0})\in\mathcal{E}_{c}(t_{0})\backslash\mathcal{\mathcal{E}}(t_{0})$
converges to $\mathcal{E}$ if $t_{f}$ is sufficiently large. The proposed
funnel synthesis aims to maximize the size of $\mathcal{E}_{c}(t_{0})$ and
minimize that of $\mathcal{E}(t)$ for all $t$ in $[t_{0},t_{f}]$.

The illustration of both the ellipsoids $\mathcal{E}(t)$ and
$\mathcal{E}_{c}(t)$ is given in Figure 1. Any solution $\eta(\cdot)$ of the
closed-loop system (3) starting at $\mathcal{E}_{c}(t_{0})$ remains in the
state funnel $\mathcal{E}_{c}(t)$ for all $t\in[t_{0},t_{f}]$ because of the
invariance condition of $\mathcal{E}_{c}$ derived in Lemma 2. Furthermore, the
solution $\eta(\cdot)$ starting at $\mathcal{E}_{c}(t_{0})$ converges to the
ellipsoid $\mathcal{E}$ if $t_{f}$ is sufficiently large because of the
attractivity of $\mathcal{E}$ given in Lemma 1. Since we use the attractivity
condition of $\mathcal{E}$ as a key property for our funnel generation, we
refer to $\mathcal{E}$ in (6) as an attractive funnel.

Additionally, with the linear feedback control $\xi=K\eta$, the condition
$\eta\in\mathcal{E}_{c}$ implies that $\xi$ is in the following ellipsoid [14,
15]:

$\displaystyle\mathcal{E}_{u}=\\{(KQK^{\top})^{\frac{1}{2}}y\mid\lVert
y\rVert_{2}\leq 1/\sqrt{c},y\in\mathbb{R}^{n_{u}}\\}.$ (10)

The set $\mathcal{E}_{u}$ represents the ellipsoid inside which the input
deviation $\xi$ remains, so we refer to $\mathcal{E}_{u}$ as an invariant
input funnel. Now, we are ready to derive the DLMI condition that guarantees
the invariant and attractive conditions.

######  Theorem 1.

Suppose that there exists $Q:[t_{0},t_{f}]\rightarrow\mathbb{S}_{++}^{n_{x}}$,
$Y:[t_{0},t_{f}]\rightarrow\mathbb{R}^{n_{u}\times n_{x}}$,
$\nu:[t_{0},t_{f}]\rightarrow\mathbb{R}_{++}$, $0<\lambda_{w}$, and $0<\alpha$
such that the following differential matrix inequality holds for all
$t\in[t_{0},t_{f}]$:

$\displaystyle H\coloneqq\left[\begin{array}[]{cccc}M-\dot{Q}&*&*&*\\\ \nu
E^{\top}&-\nu I&*&*\\\ F^{\top}&0&-\lambda_{w}I&*\\\
CQ+DY&0&G&-\nu\frac{1}{\gamma^{2}}I\end{array}\right]$ $\displaystyle\preceq
0,$ (15) $\displaystyle M\coloneqq QA^{\top}+Y^{\top}B^{\top}+AQ+BY+\alpha Q$
$\displaystyle+\lambda_{w}Q.$

Then, the Lyapunov condition (5) holds for the closed-loop system (3) with
$K=YQ^{-1}$. Thus, with $Q(t)$ and $K(t)$ satisfying the DLMI (15), the
ellipsoid $\mathcal{E}(t)$ in (6) is invariant and attractive, and
$\mathcal{E}_{c}(t)$ in (8) is invariant by Lemma 1 and Lemma 2.

###### Proof.

By definition of positive definiteness and S-procedure [12], the sufficient
condition for the Lyapunov condition (5) is that if there exists scalars
$\lambda_{p}>0$ and $\lambda_{w}>0$ such that

$\begin{aligned} \left[\begin{array}[]{ccc}\bar{M}-Q^{-1}\dot{Q}Q^{-1}&*&*\\\
E^{\top}Q^{-1}&0&*\\\
F^{\top}Q^{-1}&0&0\end{array}\right]+\lambda_{p}{\underbrace{\left[\begin{array}[]{ccc}C_{k}^{cl}&0&G_{k}\\\
0&I&0\end{array}\right]}_{\coloneqq
C_{G}}}^{\top}\left[\begin{array}[]{cc}\gamma^{2}I&0\\\
0&-I\end{array}\right]C_{G}\\\
+\lambda_{w}\left[\begin{array}[]{ccc}Q^{-1}&0&0\\\ 0&0&0\\\
0&0&-I\end{array}\right]\preceq 0,\end{aligned}$

where $\bar{M}\coloneqq A_{cl}^{\top}Q^{-1}+Q^{-1}A_{cl}+\alpha Q^{-1}$.
Applying Schur complement, and then multiplying either side by
$\text{diag}\\{Q,\lambda_{p}^{-1}I,I,I\\}$ complete the proof with
$\nu\coloneqq\lambda_{p}^{-1}$. ∎

Notice that the above differential matrix inequality (15) is linear in
$\dot{Q}$, $Q$, $Y$, and $\nu$ once $\lambda_{w}$ and $\alpha$ are fixed.

###  2.3 Feasibility of State and Input Funnels

The funnel synthesis of the proposed work aims to be not only invariant but
also feasible, so constraints on the invariant state and input funnels should
be satisfied. The feasible sets for the state and input funnels can be
described as

$\displaystyle\mathcal{X}$ $\displaystyle=\\{x\mid h_{i}(x)\leq 0,\quad
i=1,\ldots,m_{x}\\},$ $\displaystyle\mathcal{U}$ $\displaystyle=\\{u\mid
g_{j}(u)\leq 0,\quad j=1,\ldots,m_{u}\\},$

where $h_{i}:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}$ and
$g_{j}:\mathbb{R}^{n_{u}}\rightarrow\mathbb{R}$ are assumed to be at least
once differentiable (possibly nonconvex) functions. Since it is not tractable
to impose the nonconvex constraints on the ellipsoid funnel, we linearize the
above constraints around the nominal trajectory, resulting in the following
polyhedral constraints sets [3]:

$\displaystyle\mathcal{P}_{x}$ $\displaystyle=\\{x\mid(a_{i}^{h})^{\top}x\leq
b_{i}^{h},\quad i=1,\ldots,m_{x}\\},$ $\displaystyle\mathcal{P}_{u}$
$\displaystyle=\\{u\mid(a_{j}^{g})^{\top}u\leq b_{j}^{g},\quad
j=1,\ldots,m_{u}\\},$

where $(a_{i}^{h},b_{i}^{h})$ and $(a_{j}^{g},b_{j}^{g})$ are first-order
approximations of $h_{i}$ and $g_{j}$, respectively. The inclusions
$\mathcal{P}_{x}\subseteq\mathcal{X}$ and
$\mathcal{P}_{u}\subseteq\mathcal{U}$ hold if the function $h_{i}$ is a
concave function, such as ellipsoidal obstacle avoidance constraints. Here we
assume that the input constraint set $\mathcal{P}_{u}$ is bounded in
$\mathbb{R}^{n_{u}}$ to prohibit the control input from being arbitrarily
large. This assumption usually holds because the unbounded input is not
allowable in practice.

Now we aim to design $\mathcal{E}_{c}$ in (8) and $\mathcal{E}_{u}$ in (10)
with $Q$ and $K$ such that
$\\{\bar{x}\\}\oplus\mathcal{E}_{c}\subseteq\mathcal{P}_{x}$,
$\\{\bar{u}\\}\oplus\mathcal{E}_{u}\subseteq\mathcal{P}_{u}$. These conditions
could be equivalently written as [12]

$\displaystyle\lVert(Q/c)^{\frac{1}{2}}a_{i}^{h}\rVert_{2}\leq
b_{i}^{h}-(a_{i}^{h})^{\top}\bar{x},\quad i$ $\displaystyle=1,\ldots,m_{x},$
$\displaystyle\lVert(K(Q/c)K^{\top})^{\frac{1}{2}}a_{j}^{g}\rVert_{2}\leq
b_{j}^{g}-(a_{j}^{g})^{\top}\bar{u},\quad j$ $\displaystyle=1,\ldots,m_{u}.$

Squaring both sides and applying Schur complement equivalently generates

$\displaystyle 0$
$\displaystyle\preceq\left[\begin{array}[]{cc}\left(b_{i}^{h}-(a_{i}^{h})^{\top}\bar{x}\right)^{2}c&(a_{i}^{h})^{\top}Q\\\
Qa_{i}^{h}&Q\end{array}\right],$ (18) $\displaystyle 0$
$\displaystyle\preceq\left[\begin{array}[]{cc}\left(b_{j}^{g}-(a_{j}^{g})^{\top}\bar{u}\right)^{2}c&(a_{j}^{g})^{\top}Y^{\top}\\\
Ya_{j}^{g}&Q\end{array}\right],$ (21) $\displaystyle i$
$\displaystyle=1,\ldots,m_{x},\quad j=1,\ldots,m_{u}.$

The feasibility conditions (18) and (21) are linear in $Q$, $Y$, and $c$.

###  2.4 Objectives

The goal of the funnel synthesis aims to 1) maximize the size of invariant
funnel entry $\mathcal{E}_{c}(t_{0})$ from which the system can remain inside
the invariant funnel and converge to the attractive funnel, and 2) minimize
the size of the attractive funnel $\mathcal{E}(t)$ for all $t$ in
$[t_{0},t_{f}]$ for the disturbance attenuation.

First, the volume of funnel entry $\mathcal{E}_{c}(t_{0})$ is proportional to
$\det{Q(t_{0})/c(t_{0})}$ [14]. Since a log function is increasing, it is
equivalent to maximizing $\log\det(Q(t_{0})/c(t_{0}))$. It follows that

$\displaystyle\log\det(Q(t_{0})/c(t_{0}))=-n_{x}\log c(t_{0})+\log\det
Q(t_{0}).$

Hence, to maximize the volume of the funnel entry, we minimize $c(t_{0})$ and
$-\log\det Q(t_{0})$, both of which are convex functions. Second, minimizing
the volume of the set $\mathcal{E}(t)$ is equivalent to minimizing $\log\det
Q(t)$ that is a concave function. Since minimizing the concave function is a
nonconvex problem, we instead minimize the maximum radius of $\mathcal{E}(t)$
that is equal to the squared root of the maximum eigenvalue of $Q(t)$ [14].
Therefore, instead of minimizing the volume, we minimize the maximum
eigenvalue of $Q(t)$ that is a convex function.

In summary, the funnel synthesis aims to minimize a cost function $J$ given as

$\displaystyle J=w_{c}c(t_{0})-w_{Q_{0}}\log\det
Q(t_{0})+\int_{t_{0}}^{t_{f}}\bar{w}_{Q}v^{Q}(t)\text{d}t,$ (22)
$\displaystyle\text{ with }Q(t)\preceq
v^{Q}(t)I,\quad\forall\,t\in[t_{0},t_{f}],$ (23)

where $v^{Q}(t)\in\mathbb{R}_{++}$ is a slack variable introduced to minimize
the maximum eigenvalue of $Q(t)$, and
$w_{c},w_{Q_{0}},\bar{w}_{Q}\in\mathbb{R}_{++}$ are user-defined weight
parameters.

###  2.5 Continuous-time funnel synthesis problem

The continuous-time funnel synthesis problem can be formulated as follows:

$\displaystyle\underset{\leavevmode\resizebox{140.92792pt}{}{$Q(t),Y(t),c(t),\nu(t),v^{Q}(t)$}}{\operatorname{minimize}}~{}~{}$
$\displaystyle\eqref{eq:cost_function}$ (24a)
$\displaystyle\operatorname{subject~{}to}~{}~{}~{}~{}~{}$
$\displaystyle\forall\,t\in[t_{0},t_{f}],$ (24b)
$\displaystyle\eqref{eq:condition_c},\eqref{eq:DLMI_nonlinear},\eqref{eq:const_state},\eqref{eq:const_input},\eqref{eq:eq_with_slack_variable},$
(24c) $\displaystyle Q(t_{0})\succeq c(t_{0})Q_{i},$ (24d) $\displaystyle
Q(t_{f})\preceq c(t_{f})Q_{f},$ (24e)

where the matrices $Q_{i}\in\mathbb{S}_{++}^{n_{x}}$ and
$Q_{f}\in\mathbb{S}_{++}^{n_{x}}$ are constant parameters used for the
boundary conditions. These boundary conditions imply
$\mathcal{E}_{c}(t_{0})\supseteq\\{\eta\mid\eta^{\top}Q_{i}^{-1}\eta\leq 1\\}$
and $\mathcal{E}_{c}(t_{f})\subseteq\\{\eta\mid\eta^{\top}Q_{f}^{-1}\eta\leq
1\\}$.

##  3 Optimizing Funnel via Optimal Control

The problem formulated in (24) is an infinite-dimensional continuous-time
optimization problem, so it is not readily straightforward to solve it
numerically. Here we discuss a way to transform the problem into a finite-
dimensional discrete-time convex problem.

###  3.1 Changing a DLMI to a Differential Matrix Equality

In this subsection, we illustrate how the funnel synthesis problem (24) can be
interpreted as a continuous-time optimal control problem. Observe that the
DLMI in (15) can be equivalently converted into a differential matrix equality
(DME) by introducing a PSD-valued slack variable
$Z(t)\in\mathbb{S}_{+}^{n_{z}}$ with $n_{z}=n_{x}+n_{p}+n_{w}+n_{q}$ as
follows:

$\displaystyle H+\underbrace{\left[\begin{array}[]{cccc}Z^{11}&*&*&*\\\
Z^{21}&Z^{22}&*&*\\\ Z^{31}&Z^{32}&Z^{33}&*\\\
Z^{41}&Z^{42}&Z^{43}&Z^{44}\end{array}\right]}_{\coloneqq Z}$
$\displaystyle=0,\quad Z\succeq 0,$ (29)

where $H$ is defined in (15) and $Z^{ij}(t)$ have appropriate sizes for all
$i,j\in\\{1,\ldots,4\\}$. The first-row and first-column block has the
following form:

$\displaystyle\dot{Q}(t)=M(t)+Z^{11}(t).$ (30)

with $M(t)$ defined in (15). The DME (30) can be interpreted as a differential
equation for a linear dynamical system where $Q$ is a state, and $Y$ and
$Z^{11}$ are control inputs.

To derive further, we define the following vectors using the vectorization
operation:

$\displaystyle
q\coloneqq\text{vec}(Q),y\coloneqq\text{vec}(Y),z^{11}\coloneqq\text{vec}(Z^{11}),$
(31)

where the operation $\text{vec}(\cdot)$ stacks the columns to make a single
vector. Then the DME (30) can be equivalently expressed with the vector
variables in (31) as

$\displaystyle\dot{q}(t)$
$\displaystyle=A_{q}(t)q(t)+B_{q}(t)y(t)+S_{q}(t)z^{11}(t),$ (32)

with

$\displaystyle A_{q}$ $\displaystyle=(I\otimes A)+(A\otimes
I)+(\alpha+\lambda_{w})(I\otimes I),$ $\displaystyle B_{q}$
$\displaystyle=(I\otimes B)+(B\otimes I)K^{c},\quad S_{q}=(I\otimes I),$

where $K^{c}\in\mathbb{R}^{n_{x}n_{u}\times n_{x}n_{u}}$ is a commutation
matrix [16] such that $K^{c}\text{vec}(N)=\text{vec}(N^{\top})$ for any
arbitrary matrix $N\in\mathbb{R}^{n_{u}\times n_{x}}$.

###  3.2 Multiple Shooting Numerical Optimal Control

To transform (24) into the finite-dimensional discrete-time optimal control
problem, we first choose uniform time grids as
$t_{k}=t_{0}+\frac{k}{N}(t_{f}-t_{0})$ for all $k\in\mathcal{N}_{0}^{N}$. The
decision variables and the Lipschitz constant $\gamma$ at each node point are
set as $\triangle_{k}=\triangle(t_{k})$ where a placeholder $\triangle$
represents $Q,Y,Z,c,\nu$, and $\gamma$.

We apply continuous piecewise linear interpolation for $Y$, $Z$, $\nu$ and
$c^{-1}$ for each $k\in\mathcal{N}_{0}^{N-1}$ as follows:

$\displaystyle\square(t)$
$\displaystyle=\lambda_{k}^{m}(t)\square_{k}+\lambda_{k}^{p}(t)\square_{k+1},\quad\forall\,t\in[t_{k},t_{k+1}],$
$\displaystyle\lambda_{k}^{m}(t)$
$\displaystyle=\frac{t_{k+1}-t}{t_{k+1}-t_{k}},\quad\lambda_{k}^{p}(t)=\frac{t-t_{k}}{t_{k+1}-t_{k}},$
(33)

where a placeholder $\square$ stands for $Y,\nu,Z,c^{-1}$. Notice that we
apply the piecewise linear interpolation to the inverse of $c$, that is
$c^{-1}$, not $c$ itself. With this interpolation and additional conditions,
we can show that $c(t)$ satisfies the condition (9) for all
$t\in[t_{0},t_{f}]$.

######  Proposition 1.

Suppose that for each subinterval $c(t)$ satisfies

$\displaystyle c(t)$
$\displaystyle=\frac{c_{k}c_{k+1}}{\lambda_{k}^{m}(t)c_{k+1}+\lambda_{k}^{p}(t)c_{k}},\forall\,t\in[t_{k},t_{k+1}],\forall\,k\in\mathcal{N}_{0}^{N-1},$

and

$\displaystyle 0$ $\displaystyle<c_{k}\leq 1,\quad
e^{-\alpha(t_{k}-t_{0})}c_{k}\leq
c_{0},\quad\forall\,k\in\mathcal{N}_{0}^{N}.$ (34)

Then, $c(t)$ satisfies the condition (9).

###### Proof.

We want to show that
$\frac{1}{c(t)}\geq\max\\{1,e^{-\alpha(t-t_{0})}\frac{1}{c(t_{0})}\\}$ for all
$t\in[t_{0},t_{f}]$. The condition (34) implies
$1/c_{k}\geq\max\\{1,e^{-\alpha(t_{k}-t_{0})}\frac{1}{c(t_{0})}\\}$ for all
$k\in\mathcal{N}_{0}^{N}$. Notice that
$\max\\{1,e^{-\alpha(t-t_{0})}\frac{1}{c(t_{0})}\\}$ is convex in $t$, and
$1/c(t)$ is the convex combination of two points $1/c_{k}$ and $1/c_{k+1}$ for
$t\in[t_{k},t_{k+1}]$. Thus, it follows from the definition of the convex
function that
$\frac{1}{c(t)}\geq\max\\{1,e^{-\alpha(t-t_{0})}\frac{1}{c(t_{0})}\\}$ for
$t\in[t_{k},t_{k+1}]$. Since this holds for all $k\in\mathcal{N}_{0}^{N-1}$,
we complete the proof. ∎

The PD-valued function $Q(t)$ is not assumed to be piecewise linear, so $q(t)$
is not. Instead, $q(t)$ is the solution of the ordinary differential equation
in (32). Since the system (32) is linear, we can equivalently express it
through discretization with the interpolation (33) as

$\displaystyle q_{k+1}$
$\displaystyle=A_{k}^{q}q_{k}+B_{k}^{-}y_{k}+B_{k}^{+}y_{k+1}+S_{k}^{-}z_{k}^{11}+S_{k}^{+}z_{k+1}^{11},$
$\displaystyle\forall\,k\in\mathcal{N}_{0}^{N-1},$ (35)

where $q_{k}=\text{vec}(Q_{k})$, $y_{k}=\text{vec}(Y_{k})$, and
$z^{11}_{k}=\text{vec}(Z^{11}_{k})$. The matrices $A_{k}^{q}$, $B_{k}^{-}$,
$B_{k}^{+}$, $S_{k}^{-}$ and $S_{k}^{+}$ are corresponding discretized
matrices. More details in obtaining these matrices could be found in [17]. The
other blocks in (29) are imposed as

$\begin{aligned}
\nu_{k}E+Z_{k}^{21}=0,-v_{k}I+Z_{k}^{22}=0,F_{k}^{\top}+Z_{k}^{31}=0,Z_{k}^{32}=0,-\lambda_{w}I+Z_{k}^{33}=0,\\\
CQ_{k}+DY_{k}+Z_{k}^{41}=0,Z_{k}^{42}=0,G+Z_{k}^{43}=0,-\nu_{k}/\gamma_{k}^{2}I+Z_{k}^{44}=0,\end{aligned}$

(36)

for all $k$ in $\mathcal{N}_{0}^{N}$ where $F_{k}=F(t_{k})$.

###  3.3 Discrete-time convex funnel synthesis problem

The discrete-time funnel synthesis problem can be formulated as follows:

$\displaystyle\underset{\leavevmode\resizebox{155.02pt}{}{$Q_{k},Y_{k},c_{k},\nu_{k},v^{Q}_{k},\forall\,k\in\mathcal{N}_{0}^{N}$}}{\operatorname{minimize}}~{}$
$\displaystyle\begin{array}[]{l}w_{c}c_{0}-w_{Q_{0}}\log\det
Q_{0}\\\\[2.84544pt]
~{}~{}~{}~{}~{}~{}+\sum_{k=0}^{N}w_{Q}v^{Q}_{k}\end{array}$ (37c)
$\displaystyle\operatorname{subject~{}to}~{}~{}~{}~{}$ $\displaystyle Z\succeq
0,\eqref{eq:condition_c_discrete},\eqref{eq:discrete_dynamics},\eqref{eq:other_blocks},$
(37d)
$\displaystyle\eqref{eq:const_state},\eqref{eq:const_input},\eqref{eq:eq_with_slack_variable},$
(37e) $\displaystyle Q_{0}\succeq c_{0}Q_{i},\quad Q_{N}\preceq c_{N}Q_{f},$
(37f)

where the constraints (18), (21), (23) in (37e) are imposed at each $t=t_{k}$
for all $k$ in $\mathcal{N}_{0}^{N}$. The continuous cost function (22) is
discretized by lower sum between the subintervals, resulting in
$w_{Q}=\bar{w}_{Q}(t_{f}-t_{0})/N_{node}$. The optimization problem (37) is
convex with LMI constraints, resulting in a SDP problem, so that we can solve
it using any SDP solver.

###  3.4 Inter-sample constraint violation

One of the key issues in direct shooting approaches for optimal control is the
inter-sample constraint violation [18] that the constraint violation can occur
during subintervals since the constraints are enforced only at temporal nodes,
not for all time. This is also an issue for the proposed method as well as
other funnel generation approaches [2, 3, 4]. Future research will explore how
to impose relevant constraints for all time by exploiting the form of the
solutions of DME (29) [19].

##  4 Numerical Simulation

Figure 2:  The figure of the nominal trajectory and synthesized funnels
projected on position coordinates. It shows the projection of $\mathcal{E}$
(brown ellipsoid) and $\mathcal{E}_{c}$ (blue ellipsoid) at each node point.
Figure 3: The figure of input funnel (top left and right) and support value
$1/c(t)$ (bottom).

For the numerical simulation, we consider a unicycle model with addictive
disturbances written as

$\displaystyle\left[\begin{array}[]{c}\dot{r_{x}}\\\ \dot{r_{y}}\\\
\dot{\theta}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{c}u_{v}\cos\theta\\\ u_{v}\sin\theta\\\
u_{\theta}\end{array}\right]+\left[\begin{array}[]{c}0.1w_{1}\\\ 0.1w_{2}\\\
0\end{array}\right],$ (47)

where $r_{x}$, $r_{y}$, and $\theta$ are a $x$-axis position, a $y$-axis
position, are a yaw angle, respectively, and $u_{v}$ is a velocity and
$u_{\theta}$ is an angular velocity. The values $w_{1}$ and $w_{2}$ are
disturbances such that $w=[w_{1},w_{2}]^{\top},\lVert w\rVert\leq 1$. In this
model, the argument $q\in\mathbb{R}^{2}$ for the nonlinearity in (2) is
$[\theta,u_{v}]^{\top}$. We consider $N=30$ nodes evenly distributed over a
time horizon of $5$ seconds with $t_{0}=0$ and $t_{f}=5$. The boundary
parameters $Q_{i}$ and $Q_{f}$ are both $\text{diag}([0.08~{}0.08~{}0.06])$.
We consider two obstacle avoidance constraints, leading to nonconvex
constraints for the state illustrated in Fig. 2. The input constraints are
given as: $0\leq u_{v}\leq 2$ and $\lvert u_{\theta}\rvert\leq 2$. The 100
samples around the nominal trajectory are used for the local Lipschitz
constant $\gamma_{k}$ estimation for all $k$ in $\mathcal{N}_{0}^{N}$ by
following the procedure given in [3]. The weights $w_{c}$, $w_{Q_{0}}$, and
$w_{Q}$ are $10^{3}$, $0.1$, and $0.1$, respectively. The parameters $\alpha$
and $\lambda_{w}$ are $0.7$ and $0.5$, respectively. The simulation can be
reproducible by using the code at
https://github.com/taewankim1/funnel_synthesis_multiple_shooting.

The results of the proposed work are given in Fig. 2 and Fig. 3. We can see
that the generated funnel satisfies both state constraints (obstacle
avoidance) and input constraints at each node point. Also, the resulting
support value $1/c$ satisfies the constraint (9). To test the invariance and
attractivity conditions, we take a total of 100 samples, 50 from the surface
of $\mathcal{E}(t_{0})$ and 50 from that of $\mathcal{E}_{c}(t_{0})$. We
propagate each sample through the model (47) with a randomly chosen
disturbance $w$ such that $\lVert w\rVert=1$. In the bottom figure of Fig. 2,
the value of the Lyapunov function for each sample trajectory is plotted. We
can see that the invariance conditions of both $\mathcal{E}$ and
$\mathcal{E}_{c}$ hold well, and the samples starting from the surface of
$\mathcal{E}_{c}$ converge to the attractive funnel $\mathcal{E}$ due to the
attractivity condition.

##  5 Conclusions

This paper presents a funnel synthesis method for locally Lipschitz nonlinear
systems under the presence of bounded disturbances. The proposed funnel
synthesis approach aims to maximize the funnel entry while minimizing the
attractive funnel to bound the effect of the disturbances. To solve the
continuous-time funnel optimization problem having the DLMI, we apply the
direct multiple shooting optimal control method. In the numerical evaluation
with the unicycle model, the results show that the generated funnel satisfies
both invariance and feasibility properties.

## References

  * [1] S. V. Rakovic, A. R. Teel, D. Q. Mayne, and A. Astolfi, “Simple robust control invariant tubes for some classes of nonlinear discrete time systems,” in _Proceedings of the 45th IEEE Conference on Decision and Control_. IEEE, 2006, pp. 6397–6402.
  * [2] A. Majumdar and R. Tedrake, “Funnel libraries for real-time robust feedback motion planning,” _The International Journal of Robotics Research_ , vol. 36, no. 8, pp. 947–982, 2017.
  * [3] T. Reynolds, D. Malyuta, M. Mesbahi, B. Açıkmeşe, and J. M. Carson, “Funnel synthesis for the 6-dof powered descent guidance problem,” in _AIAA SciTech 2021 Forum_ , 2021, p. 0504.
  * [4] M. M. Tobenkin, I. R. Manchester, and R. Tedrake, “Invariant funnels around trajectories using sum-of-squares programming,” _IFAC Proceedings Volumes_ , vol. 44, no. 1, pp. 9218–9223, 2011.
  * [5] J. Fejlek and S. Ratschan, “Computing funnels using numerical optimization based falsifiers,” in _2022 International Conference on Robotics and Automation (ICRA)_. IEEE, 2022, pp. 4318–4324.
  * [6] T. Kim, P. Elango, and B. Acikmese, “Joint synthesis of trajectory and controlled invariant funnel for discrete-time systems with locally lipschitz nonlinearities,” _arXiv preprint arXiv:2209.03535_ , 2022.
  * [7] M. Corless, “Control of Uncertain Nonlinear Systems,” _Journal of Dynamic Systems, Measurement, and Control_ , vol. 115, no. 2B, pp. 362–372, 06 1993.
  * [8] M. V. Kothare, V. Balakrishnan, and M. Morari, “Robust constrained model predictive control using linear matrix inequalities,” _Automatica_ , vol. 32, no. 10, pp. 1361–1379, 1996.
  * [9] B. Açıkmeşe, J. M. Carson III, and D. S. Bayard, “A robust model predictive control algorithm for incrementally conic uncertain/nonlinear systems,” _International Journal of Robust and Nonlinear Control_ , vol. 21, no. 5, pp. 563–590, 2011.
  * [10] A. I. Malikov and D. I. Dubakina, “Numerical methods for solving optimization problems with differential linear matrix inequalities,” _Russian Mathematics_ , vol. 64, pp. 64–74, 2020.
  * [11] M. Diehl and S. Gros, “Numerical optimal control,” _Optimization in Engineering Center (OPTEC)_ , 2011.
  * [12] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, _Linear matrix inequalities in system and control theory_. SIAM, 1994.
  * [13] A. B. Açıkmeşe and M. Corless, “Robust tracking and disturbance rejection of bounded rate signals for uncertain/non-linear systems,” _International Journal of Control_ , vol. 76, no. 11, pp. 1129–1141, 2003\.
  * [14] S. Boyd and L. Vandenberghe, _Convex optimization_. Cambridge University Press, 2004.
  * [15] A. Kurzhanski and I. Vályi, _Ellipsoidal calculus for estimation and control_. Springer, 1997.
  * [16] J. R. Magnus and H. Neudecker, “The commutation matrix: some properties and applications,” _The annals of statistics_ , vol. 7, no. 2, pp. 381–394, 1979\.
  * [17] D. Malyuta, T. Reynolds, M. Szmuk, M. Mesbahi, B. Açıkmeşe, and J. M. Carson, “Discretization performance and accuracy analysis for the rocket powered descent guidance problem,” in _AIAA SciTech 2019 Forum_ , 2019, p. 0925.
  * [18] A. Richards and O. Turnbull, “Inter-sample avoidance in trajectory optimizers using mixed-integer linear programming,” _International Journal of Robust and Nonlinear Control_ , vol. 25, no. 4, pp. 521–526, 2015.
  * [19] L. Dieci and T. Eirola, “Positive definiteness in the numerical solution of Riccati differential equations,” _Numerische Mathematik_ , vol. 67, no. 3, pp. 303–313, 1994.


respect to a smooth $1$-section $l\times\mathbb{P}^{2}$ where
$l\cong\mathbb{P}^{1}$ is a line on $Y^{2m}$. It can be computed similarly
that the Hilbert scheme of planes on a quadric of dimension $2m+2$ has
codimension at most $6$ in $\operatorname{Gr}(3,2m+4)$. We get that
$\mathcal{Q}_{l}\to\mathbb{P}^{2}$ has a section for $m\geqslant 3$. Hence,
$Y^{2m}$ is rational for $m\geqslant 3$. Moreover, if $m=2$ and
$[\mathcal{A}^{4}]=0$, then
$\mathcal{Q}_{l}\cong\mathcal{Q}^{\prime}\to\mathbb{P}^{2}$ has a rational
section and thus $Y^{4}$ is rational. ∎

######  Remark 6.6.

We expect that Conjecture 1 holds for every flat quadric bundle of relative
even dimension under the same hypotheses. If it is true, then we do not need
to take hyperbolic reduction so that the results of quadric surface bundles
can be applied. In this case, the assumption that $Y^{2m}$ contains a
$\mathbb{P}^{m-1}$ can be removed for getting the semiorthogonal
decomposition.

##  Appendix A Non-commutative schemes

In the appendix, we give an overview of derived categories of non-commutative
schemes as in Definition A.1 generalizing Appendix D in [21] and [24, §2.2].
We will discuss the relations among
$\mathbf{D}_{\operatorname{QCoh}},\mathbf{D}(\operatorname{QCoh})$,
$\mathbf{D}(\operatorname{Coh})$ and prove the projection formula (Proposition
A.6).

######  Definition A.1.

A pair $(X,\mathcal{A}_{X})$ is a non-commutative scheme if $X$ is a
noetherian scheme, $\mathcal{A}_{X}$ is a sheaf of $\mathcal{O}_{X}$-algebras
and a quasi-coherent $\mathcal{O}_{X}$-module. A morphism

$H=(h,h_{\mathcal{A}})\colon(X,\mathcal{A}_{X})\to(Y,\mathcal{A}_{Y})$

of non-commutative schemes consists of a morphism $h\colon X\to Y$ of schemes
and a homomorphism $h_{\mathcal{A}}\colon
h^{*}\mathcal{A}_{Y}\to\mathcal{A}_{X}$ of $\mathcal{O}_{X}$-algebras.

Denote by

  * •

$\mathcal{A}_{X}^{\operatorname{op}}$ the opposite algebra of
$\mathcal{A}_{X}$,

  * •

$\operatorname{Mod}(X,\mathcal{A}_{X})$ the category of right
$\mathcal{A}_{X}$-modules,

  * •

$\operatorname{QCoh}(X,\mathcal{A}_{X})$ the category of quasi-coherent
sheaves on $X$ with right $\mathcal{A}_{X}$-module structures,

  * •

$\operatorname{Coh}(X,\mathcal{A}_{X})$ the category of coherent sheaves on
$X$ with right $\mathcal{A}_{X}$-module structures.

Further denote by

  * •

$\mathbf{D},\mathbf{D}^{-},\mathbf{D}^{\mathrm{b}}$ the unbounded, bounded
above and bounded derived categories,

  * •

$\mathbf{D}^{*}(X,\mathcal{A}_{X})$ the derived category
$\mathbf{D}^{*}(\operatorname{Coh}(X,\mathcal{A}_{X}))$ for
$*=\emptyset,-,\textrm{b}$,

  * •

$\mathbf{D}_{\operatorname{QCoh}}(X,\mathcal{A}_{X})$ (resp.,
$\mathbf{D}_{\operatorname{Coh}}(X,\mathcal{A}_{X})$) the unbounded derived
category of right $\mathcal{A}_{X}$-modules with quasi-coherent (resp.,
coherent) cohomologies.

There are pairs of adjoint functors

(A.1)
${\operatorname{QCoh}(X)}$${\operatorname{QCoh}(X,\mathcal{A}_{X}),}$$\scriptstyle{-\otimes\mathcal{A}_{X}}$$\scriptstyle{j_{Q}}$${\operatorname{Mod}(X)}$${\operatorname{Mod}(X,\mathcal{A}_{X})}$$\scriptstyle{-\otimes\mathcal{A}_{X}}$$\scriptstyle{j_{M}}$

where $j_{Q},j_{M}$ are forgetful functors, and $-\otimes\mathcal{A}_{X}$ is
left adjoint to $j_{Q}$ and $j_{M}$. When $\mathcal{A}_{X}$ is a coherent
sheaf, there is an additional pair of adjoint functors

${\operatorname{Coh}(X)}$${\operatorname{Coh}(X,\mathcal{A}_{X})}$$\scriptstyle{-\otimes\mathcal{A}_{X}}$$\scriptstyle{j}$

where $j\colon\operatorname{Coh}(X,\mathcal{A}_{X})\to\operatorname{Coh}(X)$
is the forgetful functor.

Recall that the coherator of $X$ is the functor $Q_{X}$ right adjoint to the
inclusion $\operatorname{QCoh}(X)\hookrightarrow\operatorname{Mod}(X)$. For
example, if $X$ is affine, then $Q_{X}(\mathcal{F})$ for
$\mathcal{F}\in\operatorname{Mod}(X)$ is the quasi-coherent sheaf
$\widetilde{\Gamma(X,\mathcal{F})}$ associated with $\Gamma(X,\mathcal{F})$.
Note that
$\widetilde{\Gamma(X,\mathcal{F})}\in\operatorname{QCoh}(X,\mathcal{A}_{X})$
if $\mathcal{F}\in\operatorname{Mod}(X,\mathcal{A}_{X})$. Thus, $Q_{X}$
induces a coherator functor

(A.2)
$Q_{\mathcal{A}_{X}}\colon\operatorname{Mod}(X,\mathcal{A}_{X})\to\operatorname{QCoh}(X,\mathcal{A}_{X}),$

which is right adjoint to the inclusion
$\operatorname{QCoh}(X,\mathcal{A}_{X})\hookrightarrow\operatorname{Mod}(X,\mathcal{A}_{X})$.

Let $T$ be an abelian category and let $K(T)$ be its homotopy category. Recall
that a complex $I^{\bullet}$ of objects in $T$ is called K-injective if
$\operatorname{Hom}_{K(T)}(M^{\bullet},I^{\bullet})=0$ for every acyclic
complex $M^{\bullet}$. In particular, a bounded below complex of injectives is
K-injective.

For a scheme $X$ and a complex $\mathcal{K}^{\bullet}$ of
$\mathcal{O}_{X}$-modules, $\mathcal{K}^{\bullet}$ is called K-flat if the
complex

$\operatorname{Tot}(\mathcal{F}^{\bullet}\otimes_{\mathcal{O}_{X}}\mathcal{K}^{\bullet})$

is acyclic for every acyclic complex $\mathcal{F}^{\bullet}$ of
$\mathcal{O}_{X}$-modules. In particular, a bounded above complex of flat
$\mathcal{O}_{X}$-modules is K-flat. We can define a similar notion for a non-
commutative scheme $(X,\mathcal{A}_{X})$. We say that a right
$\mathcal{A}_{X}$-module $\mathcal{K}$ is right flat if
$\mathcal{K}\otimes_{\mathcal{A}_{X}}-$ is an exact functor on
$\operatorname{Mod}(X,\mathcal{A}_{X}^{\operatorname{op}})$ and a complex
$\mathcal{K}^{\bullet}$ of right $\mathcal{A}_{X}$-modules is right K-flat if
the complex

$\operatorname{Tot}(\mathcal{K}^{\bullet}\otimes_{\mathcal{A}_{X}}\mathcal{F}^{\bullet})$

is acyclic for every acyclic complex $\mathcal{F}^{\bullet}$ of left
$\mathcal{A}_{X}$-modules. As with before, a bounded above complex of right
flat $\mathcal{A}_{X}$-modules is right K-flat. Replacing $\mathcal{A}_{X}$ by
$\mathcal{A}_{X}^{\operatorname{op}}$, we get notions for left flat and left
K-flat.

######  Lemma A.2.

(1) $\operatorname{Mod}(X,\mathcal{A}_{X})$,
$\operatorname{QCoh}(X,\mathcal{A}_{X})$ are Grothendieck abelian categories
and every complex in $\operatorname{Mod}(X,\mathcal{A}_{X})$ or
$\operatorname{QCoh}(X,\mathcal{A}_{X})$ has a K-injective resolution.

(2) For every complex $\mathcal{G}^{\bullet}$ of right
$\mathcal{A}_{X}$-modules, there exists a right K-flat complex
$\mathcal{K}^{\bullet}$ whose terms are right flat $\mathcal{A}_{X}$-modules
and a quasi-isomorphism $\mathcal{K}^{\bullet}\to\mathcal{G}^{\bullet}$ which
is termwise surjective. The same is true for complexes of left
$\mathcal{A}_{X}$-modules.

###### Proof.

(1) $\operatorname{QCoh}(X)$ is a Grothendieck abelian category. The abelian
category structure, direct sums and exact filtered colimits on
$\operatorname{QCoh}(X)$ carry over to
$\operatorname{QCoh}(X,\mathcal{A}_{X})$. By the adjointness (A.1),
$\operatorname{QCoh}(X,\mathcal{A}_{X})$ has a generator
$U\otimes\mathcal{A}_{X}$ where $U$ is a generator of
$\operatorname{QCoh}(X)$. Hence, $\operatorname{QCoh}(X,\mathcal{A}_{X})$ is a
Grothendieck abelian category. The existence of K-injective complexes follows
from [23, Tag 079P]. The proof for $\operatorname{Mod}(X,\mathcal{A}_{X})$ is
similar.

(2) We only need to prove for complexes of right $\mathcal{A}_{X}$-modules.
The proof is a modification of [23, Tag 06YF] and it suffices to show that
$\operatorname{Mod}(X,\mathcal{A}_{X})$ has enough right flat objects. We know
that $\operatorname{Mod}(X)$ has enough flat objects. For
$\mathcal{G}\in\operatorname{Mod}(X,\mathcal{A}_{X})$, there is a surjection
$\mathcal{F}\twoheadrightarrow j_{M}(\mathcal{G})$ from a flat
$\mathcal{O}_{X}$-module $\mathcal{F}$. Then its adjoint map
$\mathcal{F}\otimes\mathcal{A}_{X}\to\mathcal{G}$ is also a surjection and
$\mathcal{F}\otimes\mathcal{A}_{X}$ is a right flat $\mathcal{A}_{X}$-module.
Now $\mathcal{K}^{\bullet}$ can be constructed in the same way as loc. cit.,
which is the filtered colimit of a nice sequence of bounded above complexes of
right flat $\mathcal{A}_{X}$-modules. ∎

Thanks to the lemma above, it makes sense to talk about the right adjoint
functor $RQ_{\mathcal{A}_{X}}$.

######  Lemma A.3.

The natural functor
$\mathbf{D}(\operatorname{QCoh}(X,\mathcal{A}_{X}))\to\mathbf{D}_{\operatorname{QCoh}}(X,\mathcal{A}_{X})$
is an equivalence with quasi-inverse given by $RQ_{\mathcal{A}_{X}}$.

###### Proof.

Since the coherator functors
$Q_{\mathcal{A}_{X}}\colon\operatorname{Mod}(X,\mathcal{A}_{X})\to\operatorname{QCoh}(X,\mathcal{A}_{X})$
and $Q_{X}\colon\operatorname{Mod}(X)\to\operatorname{QCoh}(X)$ commute with
forgetful functors, the claim is a consequence of [23, Tag 09T4]. ∎

######  Lemma A.4.

When $\mathcal{A}_{X}$ is coherent, the natural functors

$\mathbf{D}^{*}(X,\mathcal{A}_{X})\to\mathbf{D}_{\operatorname{Coh}}^{*}(\operatorname{QCoh}(X,\mathcal{A}_{X}))\to\mathbf{D}_{\operatorname{Coh}}^{*}(X,\mathcal{A}_{X})$

for $*=-,\textrm{b}$ are equivalences.

###### Proof.

The equivalence of the second functor follows from Lemma A.3. For the first
functor, we will modify the proof in [23, Tag 0FDA]. We claim that if there is
a surjection $\mathcal{G}\twoheadrightarrow\mathcal{F}$ for
$\mathcal{F}\in\operatorname{Coh}(X,\mathcal{A}_{X})$ and
$\mathcal{G}\in\operatorname{QCoh}(X,\mathcal{A}_{X})$, then there is some
$\mathcal{G}^{\prime}\in\operatorname{Coh}(X,\mathcal{A}_{X})$ that surjects
onto $\mathcal{F}$. Consequently, the first functor is an equivalence by [23,
Tag 0FCL]. We know $j_{Q}(\mathcal{G})$ is a filtered union of coherent
submodules $\mathcal{G}_{i}$. Let
$\mathcal{G}_{i}^{\prime}=\operatorname{Im}(\mathcal{G}_{i}\otimes\mathcal{A}_{X}\to\mathcal{G})$
be the image of the map adjoint to the inclusion
$\mathcal{G}_{i}\hookrightarrow j_{Q}(\mathcal{G})$. Then $\mathcal{G}$ is the
filtered union of coherent $\mathcal{A}_{X}$-submodules
$\mathcal{G}_{i}^{\prime}$ and one of them will be $\mathcal{G}^{\prime}$. ∎

Given a morphism
$H=(h,h_{\mathcal{A}})\colon(X,\mathcal{A}_{X})\to(Y,\mathcal{A}_{Y})$, a
push-forward functor is defined by

$H_{*}\colon\operatorname{QCoh}(X,\mathcal{A}_{X})\to\operatorname{QCoh}(Y,\mathcal{A}_{Y})$

where as a quasi-coherent sheaf, $H_{*}\mathcal{F}$ is given by
$h_{*}\mathcal{F}$ for $\mathcal{F}\in\operatorname{QCoh}(X,\mathcal{A}_{X})$,
and a right $\mathcal{A}_{Y}$-module structure on $h_{*}\mathcal{F}$ is
induced by

$(h_{*}\mathcal{F})\otimes\mathcal{A}_{Y}\cong h_{*}(\mathcal{F}\otimes
h^{*}\mathcal{A}_{Y})\xrightarrow{h_{\mathcal{A}}}h_{*}(\mathcal{F}\otimes\mathcal{A}_{X})\to
h_{*}\mathcal{F}.$

A pull-back functor is defined by

$H^{*}\colon\operatorname{QCoh}(Y,\mathcal{A}_{Y})\to\operatorname{QCoh}(X,\mathcal{A}_{X})$

where for $\mathcal{G}\in\operatorname{QCoh}(Y,\mathcal{A}_{Y})$,

$H^{*}\mathcal{G}:=(h^{*}\mathcal{G})\otimes_{h^{*}\mathcal{A}_{Y}}\mathcal{A}_{X}\cong(h^{-1}\mathcal{G})\otimes_{h^{-1}\mathcal{A}_{Y}}\mathcal{A}_{X}.$

In the lemma below, we keep the same notations $RH_{*}$, $LH^{*}$ for the
derived push-forward and pull-back functors induced from the original ones,
respectively.

######  Lemma A.5.

Let $H=(h,h_{\mathcal{A}})\colon(X,\mathcal{A}_{X})\to(Y,\mathcal{A}_{Y})$ be
a morphism between non-commutative schemes.

(1) There exists a right derived functor

$RH_{*}\colon\mathbf{D}(\operatorname{QCoh}(X,\mathcal{A}_{X}))\to\mathbf{D}(\operatorname{QCoh}(Y,\mathcal{A}_{Y})).$

When $h$ is proper and $\mathcal{A}_{Y}$ is coherent, it induces

$RH_{*}\colon\mathbf{D}^{*}(X,\mathcal{A}_{X})\to\mathbf{D}^{*}(Y,\mathcal{A}_{Y})$

for $*=-,\textrm{b}$.

(2) There exists a left derived functor

$LH^{*}\colon\mathbf{D}(\operatorname{QCoh}(Y,\mathcal{A}_{Y}))\to\mathbf{D}(\operatorname{QCoh}(X,\mathcal{A}_{X})).$

When $\mathcal{A}_{X}$ is coherent, it induces

$LH^{*}\colon\mathbf{D}^{-}(Y,\mathcal{A}_{Y})\to\mathbf{D}^{-}(X,\mathcal{A}_{X}).$

(3) $H^{*}\dashv H_{*}$ and $LH^{*}\dashv RH_{*}$ are adjoint functors.

###### Proof.

(1) By Lemma A.2 (1), the K-injective resolutions exist for
$\mathbf{D}(\operatorname{QCoh}(X,\mathcal{A}_{X}))$ and thus the right
derived functor $RH_{*}$ can be defined. When $h$ is proper, we have an
induced functor

$RH_{*}\colon\mathbf{D}_{\operatorname{Coh}}(\operatorname{QCoh}(X,\mathcal{A}_{X}))\to\mathbf{D}_{\operatorname{Coh}}(\operatorname{QCoh}(Y,\mathcal{A}_{Y})).$

When $\mathcal{A}_{Y}$ is coherent, from Lemma A.4, we get the right derived
functor

$RH_{*}\colon\mathbf{D}^{*}(X,\mathcal{A}_{X})\to\mathbf{D}_{\operatorname{Coh}}^{*}(\operatorname{QCoh}(X,\mathcal{A}_{X}))\to\mathbf{D}_{\operatorname{Coh}}^{*}(\operatorname{QCoh}(Y,\mathcal{A}_{Y}))\cong\mathbf{D}^{*}(Y,\mathcal{A}_{Y}).$

for $*=-,\textrm{b}$.

(2) Given
$\mathcal{G}^{\bullet}\in\mathbf{D}(\operatorname{QCoh}(Y,\mathcal{A}_{Y}))$,
we define

$LH^{*}\mathcal{G}^{\bullet}:=RQ_{\mathcal{A}_{X}}(H^{*}\mathcal{K}^{\bullet})\in\mathbf{D}(\operatorname{QCoh}(X,\mathcal{A}_{X}))$

where $\mathcal{K}^{\bullet}$ is the right K-flat resolution of
$\mathcal{G}^{\bullet}$ constructed in Lemma A.2 (2) and
$RQ_{\mathcal{A}_{X}}$ is the derived coherator in Lemma A.3. Standard
arguments show that this is well-defined.

Given $\mathcal{G}\in\operatorname{Coh}(Y,\mathcal{A}_{Y})$, we claim that
$H^{*}\mathcal{G}\in\operatorname{Coh}(X,\mathcal{A}_{X})$ when
$\mathcal{A}_{X}$ is coherent. This is a local question. Assume that
$(X,\mathcal{A}_{X})\cong(\operatorname{Spec}A,\widetilde{R_{A}})$,
$(Y,\mathcal{A}_{Y})\cong(\operatorname{Spec}B,\widetilde{R_{B}})$ and
$\mathcal{G}\cong\widetilde{M}$. There is a surjection
$B^{n}\twoheadrightarrow M$ for some $n$ and it induces surjections

$R_{B}^{n}\twoheadrightarrow M,\quad R_{A}^{n}\twoheadrightarrow
M\otimes_{R_{B}}R_{A}.$

Then $M\otimes_{R_{B}}R_{A}$ is a finitely generated $A$-module because
$R_{A}$ is such.

Given $\mathcal{G}^{\bullet}\in\mathbf{D}^{-}(Y,\mathcal{A}_{Y})$, we have

$LH^{*}\mathcal{G}^{\bullet}\in\mathbf{D}_{\operatorname{Coh}}^{-}(\operatorname{QCoh}(X,\mathcal{A}_{X}))\cong\mathbf{D}^{-}(X,\mathcal{A}_{X}),$

where the equivalence is given by Lemma A.4.

(3) It suffices to prove the adjointness for $H^{*}\dashv H_{*}$. For
$\mathcal{F}\in\operatorname{Mod}(X,\mathcal{A}_{X})$ and
$\mathcal{G}\in\operatorname{Mod}(Y,\mathcal{A}_{Y})$, we have

$\displaystyle\operatorname{Hom}_{\mathcal{A}_{X}}(H^{*}\mathcal{G},\mathcal{F})$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{A}_{X}}(h^{*}\mathcal{G}\otimes_{h^{*}\mathcal{A}_{Y}}\mathcal{A}_{X},\mathcal{F})$
$\displaystyle\cong\operatorname{Hom}_{h^{*}\mathcal{A}_{Y}}(h^{*}\mathcal{G},\mathcal{F})$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{A}_{Y}}(\mathcal{G},H_{*}\mathcal{F}).$

∎

######  Proposition A.6 (Projection formula).

Let $H=(h,h_{\mathcal{A}})\colon(X,\mathcal{A}_{X})\to(Y,\mathcal{A}_{Y})$ be
a morphism of non-commutative schemes in Definition A.1.

(1) Given
$\mathcal{F}\in\mathbf{D}(\operatorname{QCoh}(X,\mathcal{A}_{X}^{\operatorname{op}}))$
and $\mathcal{G}\in\mathbf{D}(\operatorname{QCoh}(Y,\mathcal{A}_{Y}))$, there
is a natural map

(A.3) $\mathcal{G}\otimes_{\mathcal{A}_{Y}}^{\mathbb{L}}RH_{*}(\mathcal{F})\to
RH_{*}(LH^{*}(\mathcal{G})\otimes_{\mathcal{A}_{X}}^{\mathbb{L}}\mathcal{F}),$

and it is an isomorphism in $\mathbf{D}(\operatorname{QCoh}(Y))$.

(2) Assume that $h:X\to Y$ is proper and $\mathcal{A}_{X},\mathcal{A}_{Y}$ are
coherent. Given
$\mathcal{F}\in\mathbf{D}^{-}(X,\mathcal{A}_{X}^{\operatorname{op}})$ and
$\mathcal{G}\in\mathbf{D}^{-}(Y,\mathcal{A}_{Y})$, the natural map (A.3) is an
isomorphism in $\mathbf{D}^{-}(Y)$.

###### Proof.

The derived functors involved are defined in Lemma A.5 and the natural map
(A.3) is induced by the adjointness $LH^{*}\dashv RH_{*}$. The proof of the
proposition is the same as the proof of Lemma 2.5 in [24]. ∎

## References

  * [1] N. Addington.  Spinor sheaves on singular quadrics.  Proc. Amer. Math. Soc., 139(11):3867–3879, 2011.
  * [2] T. Ando.  On extremal rays of the higher-dimensional varieties.  Invent. Math., 81(2):347–357, 1985.
  * [3] A. Auel, M. Bernardara, and M. Bolognesi.  Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems.  J. Math. Pures Appl. (9), 102(1):249–291, 2014.
  * [4] A. Auel, R. Parimala, and V. Suresh.  Quadric surface bundles over surfaces.  Doc. Math., Extra vol.:(Alexander S. Merkurjev’s sixtieth birthday):31–70, 2015.
  * [5] R. Baeza.  Quadratic forms over semilocal rings.  Lecture Notes in Mathematics, Vol. 655. Springer-Verlag, Berlin-New York, 1978.
  * [6] A. Beauville.  Variétés de Prym et jacobiennes intermédiaires.  Ann. Sci. École Norm. Sup. (4), 10(3):309–391, 1977.
  * [7] W. Bichsel and M.-A. Knus.  Quadratic forms with values in line bundles.  In Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991), volume 155 of Contemp. Math., pages 293–306. Amer. Math. Soc., Providence, RI, 1994.
  * [8] A. Bodzenta and A. Bondal.  Flops and spherical functors.  Compos. Math., 158(5):1125–1187, 2022.
  * [9] A. Fujiki and S. Nakano.  Supplement to “On the inverse of monoidal transformation”.  Publ. Res. Inst. Math. Sci., 7:637–644, 1971/72.
  * [10] P. Gille and T. Szamuely.  Central simple algebras and Galois cohomology, volume 101 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2006.
  * [11] B. Hassett, A. Várilly-Alvarado, and P. Varilly.  Transcendental obstructions to weak approximation on general K3 surfaces.  Adv. Math., 228(3):1377–1404, 2011.
  * [12] Q. Jiang.  Derived categories of quot schemes of locally free quotients, i.  arXiv preprint arXiv:2107.09193, 2021.
  * [13] J. Kollár and S. Mori.  Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, 1998.  With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.
  * [14] A. Kuznetsov.  Derived categories of quadric fibrations and intersections of quadrics.  Adv. Math., 218(5):1340–1369, 2008.
  * [15] A. Kuznetsov.  Derived categories of cubic fourfolds.  In Cohomological and geometric approaches to rationality problems, volume 282 of Progr. Math., pages 219–243. Birkhäuser Boston, Boston, MA, 2010.
  * [16] A. Kuznetsov.  Base change for semiorthogonal decompositions.  Compos. Math., 147(3):852–876, 2011.
  * [17] A. Kuznetsov.  Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category.  Math. Z., 276(3-4):655–672, 2014.
  * [18] A. Kuznetsov.  Derived categories of families of sextic del pezzo surfaces.  International Mathematics Research Notices, 2017.
  * [19] A. Kuznetsov.  Quadric bundles and hyperbolic equivalence.  arXiv preprint arXiv:2108.01546, 2021.
  * [20] A. Kuznetsov and E. Shinder.  Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics.  Selecta Math. (N.S.), 24(4):3475–3500, 2018.
  * [21] A. G. Kuznetsov.  Hyperplane sections and derived categories.  Izv. Ross. Akad. Nauk Ser. Mat., 70(3):23–128, 2006.
  * [22] R. Moschetti.  The derived category of a non generic cubic fourfold containing a plane.  Math. Res. Lett., 25(5):1525–1545, 2018.
  * [23] T. Stacks project authors.  The stacks project.  https://stacks.math.columbia.edu, 2021.
  * [24] F. Xie.  Nodal quintic del pezzo threefolds and their derived categories.  arXiv preprint arXiv:2108.03186, 2021.