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Towards the theory of strong minimum in calculus of variations and optimal control: a view from variational analysis

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Abstract

The paper offers a self-contained account of the theory of first and second order necessary conditions for optimal control problems (with state constraints) based on new principles coming from variational analysis. The key element of the theory is reduction of the problem to unconstrained minimization of a Bolza-type functional with necessarily non-differentiable integrand and off-integral term. This allows to substantially shorten and simplify the proofs and to get new results not detected earlier by traditional variational techniques. This includes a totally new and easily verifiable second order necessary condition for a strong minimum in the classical problem of calculus of variations. The condition is a consequence of a new and more general second order necessary condition for optimal control problems with state constraints. Simple examples show that the new conditions may work when all known necessary conditions fail.

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Notes

  1. Nonconvex subdifferential calculus, and in particular the “extremal principle” of Kruger–Mordukhovich that can be viewed as a nonconvex extension of the separation theorem, needs closed sets and lsc functions. But the set of trajectories of a control system is typically not closed in the topology of uniform convergence and its closure contains all relaxed trajectories. Therefore the proofs based on nonconvex separation give maximum principle only for relaxed systems—see e.g. [18].

  2. The only work known for me where controllability approach is used to get second order conditions is [1]. But it is assumed in the paper that the optimal control takes value in the interior of the set of admissible controls for all t.

  3. Actually we need a bit less. It will be clear from the proofs that all we need is the existence of a Castaing representation of \(U(\cdot )\) which is a countable collection of measurable selections \(u_i(\cdot ),\ i=1,2,\ldots \) such that for almost every t the closure of the set \(\{u_1(t),u_2(t),\ldots \}\) contains U(t)

  4. In [21] a problem with variable time interval is considered plus the inequality constraints have more general structure than here.

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Authors and Affiliations

  1. Department of Mathematics, Technion, 32000, Haifa, Israel

    A. D. Ioffe

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  1. A. D. Ioffe
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Correspondence to A. D. Ioffe.

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Communicated by J. Jost.

To the memory of Professor Louis Nirenberg.

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Ioffe, A.D. Towards the theory of strong minimum in calculus of variations and optimal control: a view from variational analysis. Calc. Var. 59, 83 (2020). https://doi.org/10.1007/s00526-020-01736-2

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