Abstract
Given a nonlinear control system depending on two controls \begin{document}$u$\end{document} and \begin{document}$v$\end{document} , with dynamics affine in the (unbounded) derivative of \begin{document}$u$\end{document} and a closed target set \begin{document}$\mathcal{S}$\end{document} depending both on the state and on the control \begin{document}$u$\end{document} , we study the minimum time problem with a bound on the total variation of \begin{document}$u$\end{document} and \begin{document}$u$\end{document} constrained in a closed, convex set \begin{document}$U$\end{document} , possibly with empty interior. We revisit several concepts of generalized control and solution considered in the literature and show that they all lead to the same minimum time function \begin{document}$T$\end{document} . Then we obtain sufficient conditions for the existence of an optimal generalized trajectory-control pair and study the possibility of Lavrentiev-type gap between the minimum time in the spaces of regular (that is, absolutely continuous) and generalized controls. Finally, under a convexity assumption on the dynamics, we characterize \begin{document}$T$\end{document} as the unique lower semicontinuous solution of a regular HJ equation with degenerate state constraints.
Highlights
Let us consider the problem of minimizing the time t(x,u,v) := inf{t 0 : (x(t), u(t)) 2 S} (1)over trajectory-control pairs (x, u, v) verifying (u, v) 2 BV (IR+, U ) ⇥ M(IR+, V ), Var(u) K (K > 0); (2)X m x (t) = f (x(t), u(t), v(t)) + gi(x(t), u(t), v(t)) ui(t), t 0, (3)i=1 x(0) = x0 2 IRn, u(0) = u0 2 U, (4)where the target S ⇢ IRn ⇥ U is a closed set with compact boundary, V ⇢ IRq is a compact set and U ⇢ IRm is a closed, convex set
In this paper we focus on the minimum time problem for a general, non - commutative1 control system, where the impulsive control u is constrained in a closed, convex set U, possibly with empty interior, and the closed target S depends both on x and on u
We assume for simplicity the convexity hypothesis (H1), implying that the minimum time function T is lower semicontinuous, and characterize T as unique l.s.c. solution of a HJB equation, verifying suitable boundary conditions
Summary
Let us consider the problem of minimizing the time t(x,u,v) := inf{t 0 : (x(t), u(t)) 2 S}. They prove the existence of an optimal control and, following a level set approach, they characterize the associated capture basin Their impulsive optimization problem di↵ers from our, since they take the infimum of the time just in the subclass of space-time controls associated to rectilinear graph completions of u (see Subsection 2.2). They disregard the explicit dependence of the minimum time function on the variation bound K, which plays here an essential role (see Example 5.1).
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