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On a class of nonlinear time optimal control problems. (English) Zbl 0867.49016

Summary: We consider the minimum time optimal control problem for systems of the form \[ y'(t)=f(y(t),u(t)),\quad y(t)\in\mathbb{R}^n,\quad u(t)\in U\subset\mathbb{R}^d. \] We assume \(f(x,U)\) to be a convex set with \(C^1\) boundary for all \(x\in\mathbb{R}^n\) and the target \(\mathcal K\) to satisfy an interior sphere condition. For such problems we prove necessary and sufficient optimality conditions using the properties of the minimum time function \(T(x)\). Moreover, we give a local description of the singular set of \(T\).

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49L20 Dynamic programming in optimal control and differential games
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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