Andrei Dmitruk

Extensions of metric regularity

This article is devoted to some extensions of the metric regularity property for mappings between metric or Banach spaces. Several new concepts are investigated in a unified manner: uniform metric regularity, metric regularity along a subspace, metric multi-regularity for mappings into product spaces (when each component is perturbed independently), as well as their Lipschitz-like counterparts. The properties are characterized in terms of certain derivative-like constants. Regularity criteria are established based on a set-valued extension of a nonlocal version of the Lyusternik–Graves theorem due to Milyutin. †Dedicated to the memory of Prof. Dr. Alexender Moiseevich Rubinov; Guest Editors: Adil Bagirov and Gleb Beliakov.

On stationarity conditions in an optimal control problem with a simple contact with the phase boundary

A certain class of optimal control problems with a one-dimensional phase constraint is considered. When a trajectory contacts the phase boundary on an interval, we employ a special procedure (two-stage variation) to obtain optimality conditions in the Gamkrelidze form and then in the Dubovitskii–Milyutin form, including the sign definiteness property measure density and its jumps at junction points.

On the Proof of Pontryagin’s Maximum Principle by Means of Needle Variations

We propose a proof of the maximum principle for the general Pontryagin type optimal control problem, based on packets of needle variations. The optimal control problem is first reduced to a family of smooth finite-dimensional problems, the arguments of which are the widths of the needles in each packet, then, for each of these problems, the standard Lagrange multipliers rule is applied, and finally, the obtained family of necessary conditions is “compressed” in one universal optimality condition by using the concept of centered family of compacta.

Analysis of a Quadratic Functional with a Partly Degenerate Legendre Condition

The question of nonnegativity of a quadratic functional such that its matrix at the square of control is degenerate in part of controls is studied. After some transformation of the phase variables, the “degenerate” part of controls disappears, while its role passes to new phase variables. The obtained quadratic functional can have a non-degenerate matrix at the square of new control variables, allowing us to study its sign definiteness by standard methods.

A General Lagrange Multipliers Theorem and Related Questions

The paper deals with a general optimization problem with equality and inequality constraints in a Banach space, the latter being given by closed convex cones with nonempty interiors. A necessary optimality condition in the form of Lagrange multipliers rule is presented.

A general lagrange multipliers theorem

We consider a general optimization problem with equality and inequality constraints in a Banach space, the latter being given by closed convex cones with nonempty interiors. A necessary optimality condition in the form of Lagrange multipliers rule is presented, that is convenient for application to a wide range of optimization problems.

On two approaches to necessary conditions for an extended weak minimum in optimal control problems with state constraints

We consider a class of optimal control problems with a scalar state constraint. For a trajectory with a single boundary subarc, we first obtain, using a special technique (two-stage var iation approach), optimality conditions in the form of Gamkrelidze, and then obtain the full set of optimality conditions in the Dubovitskii-Milyutin form, including the nonnegativity of the measure density and its atoms at the junction points.

Optimal trajectories in a maximal height problem for a simplified version of the Goddard model in case of generalized media resistance function

We consider a family of the problems on maximization of the height of the vertical flight of a material point in the presence of a nonlinear friction and a constant flat gravity field under a bounded thrust and fuel expenditure. Using the maximum principle we obtain classification of trajectories (w.r.t. problem parameters) which are suspected to be optimal and check that for some classical rocket systems optimal control in our model is the classical bang-bang or bang-singular-bang one. We obtain some new types of “potentially-optimal” trajectories which should be investigated.

On the Uniform Convergence of Solutions to a Controlled System of Integral Equations with Weakly Convergent Controls

A controlled system of Volterra-type integral equations is considered which is linear with respect to the control and has integrand measurable with respect to the integration variable. It is proved that if a sequence of controls weakly converges in the space L1, then, for its members with large numbers, the system has solutions uniformly converging to the solution corresponding to the limit control.

On the uniform convergence of solutions of Volterra-type controlled systems of integral equations linear in the control

For systems of integral equations with properties cited in the title, we propose a constraint on the convergence of the controls guaranteeing the uniformconvergence of the solutions of such systems. This requirement is weaker than weak convergence. Relevant examples are given.