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1

Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints

Frankowska H.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1327-1340

EN

It is well known that for a Bolza optimal control problem under state constraints every local minimizer satisfies a constrained maximum principle which may be degenerate. In the recent years several researchers proposed sufficient conditions for its nondegeneracy, e.g. Arutyanov and Assev (1997), Rampazzo and Vinter (1999, 2000), Galbraith and Vinter (2003). In all these papers the most important assumption links dynamics of a control system with tangent cones to constraints. It is the so called inward pointing condition of control theory that is in the same spirit with the well known Slater and Managasarian-Fromowitz conditions of mathematical programming. We propose here two sufficient conditions for normality when the boundary of constraints is C1 and the end point is free. The first one applies to every nondegenerate maximum principle without any assumptions on the initial state. The second one applies to every maximum principle, but involves an additional assumption on the initial conditions.

2

Optimal synthesis via superdifferentials of value function

Frankowska H.

Control and Cybernetics

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2005

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Vol. 34, no 3

787-803

EN

We derive a differential inclusion governing the evolution of optimal trajectories to the Mayor problem. The value function is allowed to be discontinuous. This inclusion has convex compact right-hand sides.

3

Relaxing constrained control systems

Frankowska H., Rampazzo F.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 1

70-81

EN

In this paper we provide a relaxation result for control systems under both equality and inequality constraints involving the state and the control. In particular we show that the Mangasarian-Fromowitz constraint qualification allows to rewrite constrained systems as differential inclusions with locally Lipschitz right-hand side. Then Filippov-Ważewski relaxation theorem may be applied to show that ordinary solutions are dense in the set of relaxed solutions. If, besided agreeing with the above constraints, the state has to remain in a control-independent set K, then we provide a condition on the feasible velocities on the boundary of K to get a relaxation theorem.