Global controllability by nice controls. (English) Zbl 0703.93014

Grasse, Kevin A.; Sussmann, H. J.

Nonlinear controllability and optimal control, Lect. Workshop, New Brunswick/NJ (USA) 1987, Pure Appl. Math., Marcel Dekker 133, 33-79 (1990).

[For the entire collection see Zbl 0699.00040.]
Let \(\dot x=f(x,u(t))\) be a control system, where the state x resides in a finite-dimensional, differentiable manifold M and the control u takes values in a separable metric space \(\Omega\). Such a system is called globally controllable if, given any pair of states \(x_ 0,x_ 1\) in M, there exists an “admissible” control u:\({\mathbb{R}}\to \Omega\) that steers \(x_ 0\) to \(x_ 1\) along the trajectory of the system corresponding to u. An admissible control is a Lebesgue measurable mapping u:\({\mathbb{R}}\to \Omega\) for which the mapping (t,x)\(\mapsto f(x,u(t))\) satisfies local \(C^ 1\) Carathéodory conditions. Our paper deals with the following question: if a state \(x_ 1\) can be reached from a state \(x_ 0\) by a measurable admissible control, then under what conditions can \(x_ 1\) be reached from \(x_ 0\) by a “nicer” control, e.g., one that is continuous, differentiable, or polynomial. We give a precise definition of “niceness”, which appears to encompass the major classes of regular controls, and we prove that if \(x_ 1\) is normally reachable from \(x_ 0\), then \(x_ 1\) is reachable from \(x_ 0\) by a nice control. We further prove that if the system is globally controllable by measurable admissible controls, then every state is normally reachable from every other state. Consequently, a system globally controllable by measurable controls is globally controllable by nice controls.

Reviewer: Kevin A.Grasse

MSC:

93B05	Controllability
93C10	Nonlinear systems in control theory
93C15	Control/observation systems governed by ordinary differential equations
93B03	Attainable sets, reachability
49N60	Regularity of solutions in optimal control

Keywords:

control approximations; globally controllable; regular controls; normally reachable

Citations:

Zbl 0699.00040