Towards the theory of strong minimum in calculus of variations and optimal control: a view from variational analysis - Ioffe, A. D. (Calculus of variations and partial differential equations, 2020)
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.