A two-player game representation for a class of infinite horizon control problems under state constraints

Abstract

In this paper, feedback laws for a class of infinite horizon control problems under state constraints are investigated. We provide a two-player game representation for such control problems assuming time-dependent dynamics and Lagrangian and the set constraints merely compact. Using viability results recently investigated for state constrained problems in an infinite horizon setting, we extend some known results for the linear-quadratic regulator problem to a class of control problems with nonlinear dynamics in the state and affine in the control. Feedback laws are obtained under suitable controllability assumptions.

Appendix

Lemma A.1

Adopt the assumptions of Lemma 4.6, and assume that A and B are locally absolutely continuous and $B\in L^\infty (t,T;\mathbb {R}^{n\times m})$. Then, there exists $\beta >0$ such that for all $x\in \Omega$ and $\sigma >0$ we can find $\xi ^\sigma (\cdot )$ feasible for (3.2) on [t, T], with $\xi ^\sigma (t)=x$, satisfying

$$\begin{aligned} \begin{aligned} \begin{aligned} \left\| \xi -\xi ^\sigma \right\| _{\infty ,[t,T]}\le \beta \sigma ,\qquad \xi ^\sigma (\cdot )\subset \text{ int } \Omega . \end{aligned} \end{aligned} \end{aligned}$$

(A.1)

Proof

We take the same notation as in the proof of Lemma 4.6. We show the following claim: There exist $\varepsilon >0$ and $\eta >0$ satisfying for all $(s,x) \in [t, T] \times (\partial \Omega +\eta \mathbb {B}) \cap \Omega$ and all $y \in (x+\varepsilon \mathbb {B}) \cap \Omega$

$$\begin{aligned} \begin{aligned} y+[0, \varepsilon ]({\hat{f}}(s,x)+\varepsilon \mathbb {B}) \subset \Omega . \end{aligned} \end{aligned}$$

(A.2)

Notice that for any $(s,x) \in [t, T] \times \partial \Omega$ and from the characterization of the interior of the Clarke tangent cone (cfr [3]), we can find $\varepsilon \in (0,1)$ such that $y+[0, \varepsilon ]({\hat{f}}(s,x)+2 \varepsilon \mathbb {B}) \subset \Omega$ for all $y \in (x+2 \varepsilon \mathbb {B}) \cap \Omega$. Now, take any ${\tilde{y}} \in \left( \tilde{x}+\varepsilon \mathbb {B}\right) \cap \Omega .$ Then, since $\tilde{x}+\varepsilon \mathbb {B} \subset x+2 \varepsilon \mathbb {B}$ and $|{\hat{f}}({\tilde{t}},{\tilde{x}}) |\le |{\hat{f}}(s,x)|+\varepsilon ,$ we may conclude ${\tilde{y}}+[0, \varepsilon ]\left( {\hat{f}}(\tilde{t},{\tilde{x}})+\varepsilon \mathbb {B}\right) \subset \Omega$ for all ${\tilde{y}} \in \left( {\tilde{x}}+\varepsilon \mathbb {B}\right) \cap \Omega$. So, we have shown that for any $(s,x) \in [t, T] \times \partial \Omega$ there exist $\varepsilon _{s,x} \in (0,1)$ and $\delta _{s,x} \in \left( 0, \varepsilon _{s,x}\right]$ such that, given any $\left( {\tilde{t}}, {\tilde{x}}\right) \in \left( (s,x)+\delta _{s,x} \mathbb {B}\right) \cap ([t, T] \times \Omega ),$

$$\begin{aligned} \left\{ {\tilde{y}}+\left[ 0, \varepsilon _{s,x}\right] \left( {\hat{f}}({\tilde{t}},{\tilde{x}})+\varepsilon _{s,x}\mathbb {B}\right) \,:\,y \in \left( {\tilde{x}}+\varepsilon _{s,x}\mathbb {B}\right) \cap \Omega \right\} \subset \Omega . \end{aligned}$$

Using a compactness argument, we conclude that there exist $\left( t_{i}, x_{i}\right) \in [t, T] \times \partial \Omega$ and $0<\delta _i<\varepsilon _i$, for $i\in \{1,...,N\}$, such that $[t, T] \times \partial \Omega \subset \bigcup _{i=1}^{N}\left( \left( t_{i}, x_{i}\right) +\delta _{i} \text {int}\,\mathbb {B}\right)$, and, for any $\left( {\tilde{t}}, \tilde{x}\right) \in \left( \left( t_{i}, x_{i}\right) +\delta _{i} \mathbb {B}\right) \cap ([t, T] \times \Omega ),$

$$\begin{aligned} {\tilde{y}}+\left[ 0, \varepsilon _{i}\right] ({\hat{f}}({\tilde{t}},\tilde{x})+\varepsilon _{i} \mathbb {B}) \subset \Omega \quad \forall {\tilde{y}} \in \left( {\tilde{x}}+\varepsilon _{i} \mathbb {B}\right) \cap \Omega . \end{aligned}$$

Notice also that there exists $\eta \in \left( 0, \min _{i} \delta _{i}\right)$ satisfying $[t, T] \times (\partial \Omega +\eta \mathbb {B}) \subset \bigcup _{i=1}^{N}((t_{i}, x_{i})$ $+\delta _{i} \text {int}\,{B})$. (Otherwise we could find a sequence of points $\left( s_{j}, y_{j}\right) \notin \bigcup _{i=1}^{N}\left( \left( t_{i}, x_{i}\right) +\delta _{i} \mathbb {B}\right)$ such that $\left( s_{j}, y_{j}\right) \rightarrow (s, y) \in [t, T] \times \partial \Omega$.) The claim (A.2) just follows taking $\varepsilon =\min _{i} \varepsilon _{i}$. Consider now the following differential inclusion

$$\begin{aligned} \begin{aligned} \xi '(s)\in G(s,\xi (s)) \quad s\in [t,T]\text { a.e.},\quad \xi (t)=x, \end{aligned} \end{aligned}$$

(A.3)

where $G(s,x)\doteq \{ f_0(s,x)+f_1(s,x)u\,:\, |u|\le \left\| B^\top P \right\| _{\infty ,[t,T]}\left\| h \right\| _{\infty ,\Omega }\}$ and $f_0,f_1$ are as in (4.1). Notice that ${\hat{f}}(s,x)\in G(s,x)$ for any $(s,x)\in [t,T]\times \mathbb {R}^n$ and the trajectory $\xi (\cdot )$ is a solution of (A.3) with $u(s)=-B^\top (s)P(s)h(\xi (s))$. Moreover, $\sup \{|v|\,:\, v\in G(s,x), s\in [t,T],x\in (\partial \Omega +\mathbb {B})\} =M<\infty$ and there exists $\lambda \in L^1([t,T];{\mathbb {R}}^+)$ such that $G(s,x)\subset G(s',x)+\int _s^{s'} \lambda (\tau )d\tau$ for all $(t\le s<s'\le T)$ and $x\in \Omega$. So, arguing in analogous way as in [7, Theorem 1] and using (A.2), we conclude that there exists $\beta >0$ (depending on the time interval [t, T]) such that for any $x\in \Omega$ and any $\sigma >0$ there is a feasible trajectory $\xi ^\sigma (\cdot )$, solving (A.3) on [t, T] and starting from x, that satisfies (A.1). Hence, the conclusion follows by applying the measurable selection theorem. $\square$