Transversality condition and optimization of higher order ordinary differential inclusions

In the present paper, a Bolza problem of optimal control theory with a fixed time interval given by convex and nonconvex second-order differential inclusions (P_H) is studied. Our main goal is to derive sufficient optimality conditions for Cauchy problem of sth-order differential inclusions. The sufficient conditions including distinctive transversality condition are proved incorporating the Euler–Lagrange and Hamiltonian type inclusions. The basic concepts involved in obtaining optimality conditions are the locally adjoint mappings. Furthermore, the application of these results is demonstrated by solving the problems with third-order differential inclusions.     

Approximation and optimization of discrete and differential inclusions described by inequality constraints

In the first part of this article optimization of polyhedral discrete and differential inclusions is considered, the problem is reduced to convex minimization problem and the necessary and sufficient condition for optimality is derived. The optimality conditions for polyhedral differential inclusions based on discrete-approximation problem according to continuous problems are formulated. In particular, boundedness of the set of adjoint discrete solutions and upper semi-continuity of the locally adjoint mapping are proved. In the second part of this article an optimization problem described by convex inequality constraint is studied. By using the equivalence theorem concerning the subdifferential calculus and approximating method necessary and sufficient condition for discrete-approximation problem with inequality constraint is established.     

The optimality principle for discrete and first order partial differential inclusions

Necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of the apparatus of locally conjugate mappings and local tents.     

Optimization of higher order differential inclusions with initial value problem

This paper concerns the sufficient conditions of optimality for initial value problem with higher order differential inclusions (HODIs) and free endpoint constraints. Formulation of the transversality conditions plays a substantial role in the next investigations without which hardly any necessary or sufficient conditions would be obtained. In terms of Euler–Lagrange and Hamiltonian forms the sufficient conditions of optimality both for convex and “non-convex” HODIs are based on the apparatus of locally adjoint mappings. Moreover, by applying the main result to a Bolza problem described by a polynomial differential operator with constant coefficients in terms of the adjoint differential operator the sufficient condition of optimality is obtained.     

Approximation and optimization of Darboux type differential inclusions with set-valued boundary conditions

The present paper is devoted to an optimal control problem given by hyperbolic discrete (P _D ) and differential inclusions (P _C ) of generalized Darboux type and ordinary discrete inclusions. The results are extended to non-convex problems. An approach concerning necessary and sufficient conditions for optimality is proposed. In order to formulate sufficient conditions of optimality for problem (P _C ) the approximation method is used. Formulation of these conditions is based on locally adjoint mappings. Moreover for construction of adjoint partial differential inclusions the equivalence theorems of locally adjoint mappings are proved. One example with homogeneous boundary conditions is considered.     

Approximation of non-monotone multivalued differential inclusions

An approximation of non-monotone, multivalued differential inclusions,the so-called hemivariationa] inequalities, is presented. Arelation between the approximation problem and the problem offinding a substationary point of an approximation of the correspondingpotential function is also established. This relation makesit possible to solve the approximated problem by using nonsmooth,non-convex optimization methods. Some numerical experimentsare also presented.     

Approximation of slow solutions to differential inclusions

To approach a viable solution of a differential inclusion, i.e., staying at any time in a closed convexK, a sufficient condition is given implying the convergence of an approximation sequence defined from the Euler or Runge-Kutta methods applied to a selection process which corresponds to the slowsolution concept. WhenK is smooth, the convergence condition is satisfied. This proves that the method is implementable on a computer for solving, for instance, differentiable equations with a noncontinuous right-hand side. Since the usual best approximation operator is difficult to implement, we introduce a class of quasi-projectors much more suitable for computation.     

Directionally differentiable multiobjective optimization involving discrete inclusions

This paper is devoted to multiobjective optimization problems involving discrete inclusions. The objective functions are assumed to be directionally differentiable and the domination structure is defined by a closed convex cone. The directional derivatives are not assumed to be linear or convex. Several concepts of optimal solutions are analyzed, and the corresponding necessary conditions are obtained as well in maximum principle form. As an application of the main results, a maximum principle is also derived for multiobjective optimization with extremalvalue fucctions involving discrete inclusions.The authors are indebted to the referee for detailed comments.The paper was written while the second author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.     

Filippov Lemma for certain second order differential inclusions

In the paper we give an analogue of the Filippov Lemma for the second order differential inclusions with the initial conditions y(0) = 0, y??(0) = 0, where the matrix A ?? ? ^d×d and multifunction is Lipschitz continuous in y with a t-independent constant l. The main result is the following: Assume that F is measurable in t and integrably bounded. Let y ₀ ?? W ^2,1 be an arbitrary function fulfilling the above initial conditions and such that where p ₀ ?? L ¹[0, 1]. Then there exists a solution y ?? W ^2,1 to the above differential inclusions such that a.e. in [0, 1], .     

Oscillation of higher order delay differential equations

A sufficient condition was obtained for oscillation of all solutions of theodd-order delay differential equation $$x^{(n)} (t) + \sum\limits_{i = 1}^m {p_i (t)} x(t - \sigma _{_i } ) = 0,$$ wherep _i (t) are non-negative real valued continuous function in [T ∞] for someT≥0 and σ_i,∈(0, ∞)(i = 1,2,…,m). In particular, forp _i (t) =p _i ∈(0, ∞) andn > 1 the result reduces to $$\frac{1}{m}\left( {\sum\limits_{i = 1}^m {(p_i \sigma _i^m )^{1/2} } } \right)^2 > (n - 2)!\frac{{(n)^n }}{e},$$ implies that every solution of (*) oscillates. This result supplements forn > 1 to a similar result proved by Ladaset al [J. Diff. Equn.,42 (1982) 134–152] which was proved for the casen = 1.     

Approximation of analytic sets by Nash tangents of higher order

The aim of this paper is to show that for every locally analytic subset X of C ^m and every there exist a neighborhood V of a in C ^m and a sequence of Nash subsets of V converging to such that X _ν and X satisfy a certain condition for tangency of order ν. Next it is shown that this condition implies that for sufficiently large ν the multiplicities of X _ν and X at a are equal.      

On boundary-value problems for second order perturbed differential inclusions

This article studies the existence of solutions to boundary-value problems for second order multi-valued perturbed differential inclusions under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions is also obtained under certain monotonicity conditions and the weaker nonconvexity conditions for multi-valued functions.     

Existence results for second order impulsive functional differential inclusions

In this paper, the existence of solution on a compact interval to second order impulsive functional differential inclusions is investigated. Several new results are obtained by using suitable fixed point theorem.     

Necessary and sufficient conditions for discrete and differential inclusions of elliptic type

This paper deals for the first time with the Dirichlet problem for discrete (PD), discrete approximation problem on a uniform grid and differential (PC) inclusions of elliptic type. In the form of Euler-Lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings. The results obtained are generalized to the multidimensional case with a second order elliptic operator.     

Nonlinear boundary value problems for second order differential inclusions

In this paper, we study a class of nonlinear value boundary problems for second order differential inclusions with nonlinear perturbations, which satisfy the generalized Hartman condition weaker than that considered in some papers. Using techniques from multivalued analysis, theory of monotone operators and fixed points, we prove the existence of solutions in both “convex” and “nonconvex” cases. Our framework can be incorporated with Dirichlet, Neumann, and mixed boundary problems.