State constrained $L^\infty$ optimal control problems interpreted as differential games
Piernicola Bettiol
We consider state constrained optimal control problems in which the cost to minimize comprises an $L^\infty$ functional, i.e. the maximum of a running cost along the trajectories. In absence of state constraints, a new approach has been suggested by a recent paper [9]. The main purpose of the present paper is to extend this approach and the related results to state constrained $L^\infty$ optimal control problems. More precisely, using the $(L^\infty, L^1)$-duality, the reference optimal control problem can be seen as a static differential game, in which an extra variable is introduced and plays the role of an opponent player who wants to maximize the cost. Under appropriate assumptions and employing suitable Filippov's type results, this static game turns out to be equivalent to the corresponding dynamic differential game, whose (upper) value function is the unique viscosity solution to a constrained boundary value problem, which involves a Hamilton-Jacobi equation with a continuous Hamiltonian.
keywords: Minimax problems differential games viscosity solutions.
Estimates on trajectories in a closed set with corners for $(t,x)$ dependent data
Piernicola Bettiol Richard Vinter
Estimates on the distance of a given process from the set of processes that satisfy a specified state constraint in terms of the state constraint violation are important analytical tools in state constrained optimal control theory; they have been employed to ensure the validity of the Maximum Principle in normal form, to establish regularity properties of the value function, to justify interpreting the value function as a unique solution of the Hamilton-Jacobi equation, and for other purposes. A range of estimates are required, which differ according the metrics used to measure the `distance' and the modulus $\theta(h)$ of state constraint violation $h$ in terms of which the estimates are expressed. Recent research has shown that simple linear estimates are valid when the state constraint set $A$ has smooth boundary, but do not generalize to a setting in which the boundary of $A$ has corners. Indeed, for a velocity set $F$ which does not depend on $(t,x)$ and for state constraints taking the form of the intersection of two closed spaces (the simplest case of a boundary with corners), the best distance estimates we can hope for, involving the $W^{1,1,}$ metric on state trajectories, is a super-linear estimate expressed in terms of the $h|\log(h)|$ modulus. But, distance estimates involving the $h|\log (h)|$ modulus are not in general valid when the velocity set $F(.,x)$ is required merely to be continuous, while not even distance estimates involving the weaker, Hölder modulus $h^{\alpha}$ (with $\alpha$ arbitrarily small) are in general valid, when $F(.,x)$ is allowed to be discontinuous. This paper concerns the validity of distance estimates when the velocity set $F(t,x)$ is $(t,x)$-dependent and satisfy standard hypotheses on the velocity set (linear growth, Lipschitz $x$-dependence and an inward pointing condition). Hypotheses are identified for the validity of distance estimates, involving both the $h|\log(h)|$ and linear moduli, within the framework of control systems described by a controlled differential equation and state constraint sets having a functional inequality representation.
keywords: Control systems optimal control. state constraints
Lipschitz regularity of solution map of control systems with multiple state constraints
Piernicola Bettiol Hélène Frankowska
Consider a closed subset $K \subset \mathbb{R}^n$ and $f:[0,T]\times \mathbb{R}^n\times U \to \mathbb{R}^n$, where $U$ is a complete separable metric space. We associate to these data the control system under a state constraint \begin{equation*}\label{dm400} \left \{ \begin{array}{lll} x'(t) &=&f(t,x(t),u(t)), \; \; u(t)\in U \quad\; \mbox{ a.e. in }\; [0,T] \\ x(t) & \in & K \quad\; \mbox{ for all }\; t \in [0,T]\\ x(0)& = &x_0 . \end{array} \right. \end{equation*} When the boundary of $K$ is smooth, then an inward pointing condition guarantees that under standard assumptions on $f$ (measurable in $t$, Lipschitz in $x$, continuous in $u$) the sets of solutions to the above system depend on the initial state $x_0$ in a Lipschitz way. This follows from the so-called Neighboring Feasible Trajectories (NFT) theorems. Some recent counterexamples imply that NFT theorems are not valid when $f$ is discontinuous in time and $K$ is a finite intersection of sets with smooth boundaries, that is in the presence of multiple state constraints.
    In this paper we prove that for multiple state constraints the inward pointing condition yields local Lipschitz dependence of solution sets on the initial states from the interior of $K$. Furthermore we relax the usual inward pointing condition. The novelty of our approach lies in an application of a generalized inverse mapping theorem to investigate feasible solutions of control systems. Our results also imply a viability theorem without convexity of right-hand sides for initial states taken in the interior of $K$.
keywords: inverse mapping theorem. state constraints metric regularity control system Lipschitz dependence on initial conditions Differential inclusion

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