Abstract
Time optimal control problems for some non-smooth systems in general form are considered. The non-smoothness is caused by singularity. It is proved that Pontryagin's maximum principle holds for at least one optimal relaxed control. Thus, Pontryagin's maximum principle holds when the optimal classical control is a unique optimal relaxed control. By constructing an auxiliary controlled system which admits the original optimal classical control as its unique optimal relaxed control, one get a chance to get Pontryagin's maximum principle for the original optimal classical control. Existence results are also considered.
Highlights
Consider the following non-smooth controlled system: dy(t) dt= f (t, y(t), u(t)), y(0) = y0, t > 0, (1.1)where the state function y(·) is a vector-valued function, while the control function u(·) values in some metric space U .Hongwei Lou, Junjie Wen and Yashan XuIn this paper, the non-smoothness means that f may singular at target set
While in [9], the “target set” is {∞}, and f is unbounded near the target. We observed that both optimal quenching time problems and optimal blowup time problems can be transformed to time optimal control problems for some non-smooth systems (see Problem (T)) by some suitable state transform
To study the existence of optimal control, we introduce the relaxed control which can help to study the necessary conditions of optimal controls
Summary
Where the state function y(·) is a vector-valued function, while the control function u(·) values in some metric space U. While in [9], the “target set” is {∞}, and f is unbounded near the target We observed that both optimal quenching time problems and optimal blowup time problems can be transformed to time optimal control problems for some non-smooth systems (see Problem (T)) by some suitable state transform. We get some existence results but only establish Pontryagin’s maximum principle for one of relaxed optimal controls for general systems. Any triple (w∗, y∗(·), σ∗(·)) ∈ RPad satisfying (2.5) is called an optimal relaxed triple of Problem (R)/Problem (T). We give the following lemma, which concerns the continuous dependence of the solutions of (2.3) with respect to the relaxed controls in the meaning of convergence in R(U ).
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