D. Yu. Karamzin

Necessary optimality conditions in an abnormal optimization problem with equality constraints

An abnormal minimization problem with equality constraints and a finite-dimensional image is examined. Second-order necessary conditions for this problem are given that strengthen previously known results.

Maximum principle in problems with mixed constraints under weak assumptions of regularity

In the present work, optimal control problems with mixed constraints are investigated. A novel weakening of the conventional regularity assumptions on mixed constraints is introduced. A maximum principle is derived in which the maximum condition is of nonstandard type: the maximum is taken over the closure of the set of regular points, but not over the whole feasible set.

Necessary optimality conditions for abnormal problems with geometric constraints

The abnormal minimization problem with a finite-dimensional image and geometric constraints is examined. In particular, inequality constraints are included. Second-order necessary conditions for this problem are established that strengthen previously known results.

Conditions for the absence of jumps of the solution to the adjoint system of the maximum principle for optimal control problems with state constraints

Properties of Lagrange multipliers from the Pontryagin maximum principle for problems with state constraints are investigated. Sufficient conditions for the continuity of the solution of the adjoint equation depending on how the extremal trajectory approaches the state constraint boundary are obtained. The proof uses the notion of closure with respect to measure of a Lebesgue measurable function and the Carathéodory theorem.

The dines theorem and some other properties of quadratic mappings

Real homogeneous quadratic mappings from Rn to R2 are examined. It is known that the image of such a mapping is always convex. A proof of the convexity of the image based on the quadratic extremum principle is given. The following fact is noted: If the quadratic mapping Q is surjective and n > 2 + dimkerQ, then there exists a regular zero of Q. A certain criterion of the linear dependence of quadratic forms is also stated.

State Constraints in Impulsive Control Problems: Gamkrelidze-Like Conditions of Optimality

An impulsive control problem with state constraints is considered. A Pontryagin maximum principle in the framework of R.V. Gamkrelidze is derived, being its proof based on a certain penalization technique and on the application of Ekeland’s variational principle. This approach is distinct from the more usual ones in Impulsive Control theory based on a reduction to a conventional control problem and exhibits the advantage of allowing to address problems with dynamics which are merely measurable in the time variable. Controllability assumptions to ensure the non-degeneracy of the conditions are provided in the impulsive control context. An example demonstrating the significance of the conditions is given.

Maximum principle in an optimal control problem with equality state constraints

We consider an optimal control problem with equality state constraints. We prove nondegenerate necessary optimality conditions in the form of the Pontryaginmaximum principle.

On some properties of the shortest curve in a compound domain

We consider a state space domain defined by a regular system of equality and inequality constraints. We study the properties of the shortest curve, that is, the curve that has the minimum length of all smooth curves joining two given points of the domain and lying entirely in the domain. If inequality constraints are absent, then the shortest curve is a geodesic. We show that the shortest curve is a function of the class W2,∞, derive the equation of the shortest curve, and study some other properties of this curve.

Investigation of regularity conditions in optimal control problems with geometric mixed constraints

We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix.

Properties of extremals in optimal control problems with state constraints

An optimal control problem with state constraints is considered. Some properties of extremals to the Pontryagin maximum principle are studied. It is shown that, from the conditions of the maximum principle, it follows that the extended Hamiltonian is a Lipschitz function along the extremal and its total time derivative coincides with its partial derivative with respect to time.