### 2-Regularity and 2-Normality Conditions for Systems With Impulsive Controls

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We consider a class of age-structured control problems with nonlocal dynamics and boundary conditions. For these problems we suggest Arrow-type sufficient conditions for optimality of problems defined on finite as well as infinite time intervals. We examine some models as illustrations (optimal education and optimal offence control problems).

We construct explicitly piecewise affine mappings u:ℝ ⁿ → ℝ ⁿ with affine boundary data satisfying the constraint div u = 0. As an application of the construction we give short and direct proofs of the main approximation lemmas with constraints in convex integration theory. Our approach provides direct proofs avoiding approximation by smooth mappings and works in all dimensions n ≥ 2. After a slight modification of our construction, the constraint div u = 0 can be turned into det Du = 1, giving...

The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set R(t) ⊂ ℝ2, described as the reachable set for a differential inclusion. To restrict its growth, a barrier Γ can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification of blocking arcs. Moreover, we derive local and...

The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set R(t) ⊂ ℝ2, described as the reachable set for a differential inclusion. To restrict its growth, a barrier Γ can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification...

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula...

We prove an approximation theorem in generalized Sobolev spaces with variable exponent ${W}^{1,p\left(\xb7\right)}\left(\Omega \right)$ and we give an application of this approximation result to a necessary condition in the calculus of variations.