Optimal control problem for impulsive systems with integral boundary conditions

Abstract

In the present work the optimal control problem is considered, when the state of the system is described by the impulsive differential equations with integral boundary conditions. Applying the Banach contraction principle the existence and uniqueness of solution is proved for the corresponding boundary problem by the fixed admissible control. The first and second variation of the functional is calculated. Various necessary conditions of optimality of the first and second order are obtained by the help of the variation of the controls.

On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

“…is a set of complex coefficients. Then, → 0 as → ∞, for all ∈ N from (20), so that lim → ∞ (P ( )) = {0}(∈ ). If there are some multiple eigenvalues, with all being of finite multiplicity since the operator : → is compact, the above expression may be reformulated with projections on the finite-dimensional eigenspaces associated to each of the eventually repeated eigenvalues leading to…”

“…Hilbert spaces for the formulation of equilibrium points, stability, controllability [16,18,19], boundedness, and square integrability (or summability in the discrete formalism) of the solution in the framework of square-integrable (or squaresummable) control and output functions are of relevant importance in signal processing and control theory and in general formulations of dynamic systems, in general. See, for instance, [1,2,7,9,16,17,19,20] and the references therein. Two examples with the use of the above formalism to dynamic systems and control issues are now discussed in detail.…”

Existence and uniqueness of the solutions to impulsive nonlinear integro-differential equations with nonlocal boundary conditions

“…Today, there exist a great number of works devoted to ordinary impulsive differential and integro-differential equations with nonlocal boundary conditions in which the theorem on the existence of solutions are proved for different types of nonlocal conditions [3][4][5][6][7][8]11,13,[18][19][20][21][22].…”

Existence and uniqueness of solutions for nonlinear impulsive differential equations with three-point boundary conditions

“…Some authors have produced an extensive portfolio of results on differential equations undergoing impulse effects. The existence questions for impulsive differential equations have been studied in [10][11][12][13][14][15][16][17][18][19] and references therein. Three-point boundary value problems for ordinary differential equations are also studied in recent years.…”