This paper considers an optimal control problem for a class of controlled hybrid dynamical systems (HDSs) with prescribed switchings. By using Ekeland’s variational principle and a matrix cost functional, a minimum principle for HDSs is derived, which provides a necessary condition of the aforementioned problem. The results given in this paper include both pure continuous systems and pure discrete-time systems as special cases.

In this paper, an efficient computation method is developed for solving a general class of minmax optimal control problems, where the minimum deviation from the violation of the continuous state inequality constraints is maximized. The constraint transcription method is used to construct a smooth approximate function for each of the continuous state inequality constraints. We then obtain an approximate optimal control problem with the integral of the summation of these smooth approximate functions as its cost function. A necessary condition and a sufficient condition are derived showing the relationship between the original problem and the smooth approximate problem. We then construct a violation function from the solution of the smooth approximate optimal control problem and the original continuous state inequality constraints in such a way that the optimal control of the minmax problem is equivalent to the largest root of the violation function, and hence can be solved by the bisection search method. The control parametrization and a time scaling transform are applied to these optimal control problems. We then consider two practical problems: the obstacle avoidance optimal control problem and the abort landing of an aircraft in a windshear downburst.

Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory Appl. (2006), Art. ID 95453, 10pp] and Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math.13(6) (2009)] studied an implicit viscosity method for approximating solutions of variational inequalities by solving hierarchical fixed point problems. The approximate solutions are a net (xs,t) of two parameters s,t∈(0,1), and under certain conditions, the iterated lim t→0lim s→0xs,t exists in the norm topology. Moudafi, Maingé and Xu stated the problem of convergence of (xs,t) as (s,t)→(0,0) jointly in the norm topology. In this paper we further study the behaviour of the net (xs,t); in particular, we give a negative answer to this problem.

The deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

In this paper we extend results of Inoan and Kolumban on pseudomonotone set-valued mappings to topological vector spaces. An application is made to a variational inequality problem.

The existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via sub- and supersolution methods as well as variational techniques for nonsmooth functions.

In this paper, we introduce and consider a new class of complementarity problems, which are called the generalized mixed quasi-complementarity problems in a topological vector space. We show that the generalized mixed quasi-complementarity problems are equivalent to the generalized mixed quasi variational inequalities. Using a new type of KKM mapping theorem, we study the existence of a solution of the generalized mixed quasi-variational inequalities and generalized mixed quasi-complementarity problems. Several special cases are also discussed. The results obtained in this paper can be viewed as extension and generalization of the previously known results.

In this paper, a new approach to a characterization of solvability of a nonlinear nonsmooth multiobjective programming problem with inequality constraints is introduced. A family of η-approximated vector optimization problems is constructed by a modification of the objective and the constraint functions in the original nonsmooth multiobjective programming problem. The connection between (weak) efficient points in the original nonsmooth multiobjective programming problem and its equivalent η-approximated vector optimization problems is established under V-invexity. It turns out that, in most cases, solvability of a nonlinear nonsmooth multiobjective programming problem can be characterized by solvability of differentiable vector optimization problems.

The problem of optimal dividends paid until absorbtion at zero is considered for a rather general diffusion model. With each dividend payment there is a proportional cost and a fixed cost. It is shown that there can be essentially three different solutions depending on the model parameters and the costs. (i) Whenever assets reach a barrier y*, they are reduced to y* - δ* through a dividend payment, and the process continues. (ii) Whenever assets reach a barrier y*, everything is paid out as dividends and the process terminates. (iii) There is no optimal policy, but the value function is approximated by policies of one of the two above forms for increasing barriers. A method to numerically find the optimal policy (if it exists) is presented and numerical examples are given.

Assume that a Banach space has a Fréchet differentiable and locally uniformly convex norm. We show that the reflexive property of the Banach space is not only sufficient, but also a necessary condition for the fulfillment of the proximal extremal principle in nonsmooth analysis.

The existence of strictly feasible points is shown to be equivalent to the boundedness of solution sets of generalized complementarity problems with stably pseudomonotone mappings. This generalizes some known results in the literature established for complementarity problems with monotone mappings.

We give a multidirectional mean value inequality with second order information. This result extends the classical Clarke-Ledyaev's inequality to the second order. As application, we give the uniqueness of viscosity solution of second order Hamilton-Jacobi equations in finite dimensions.

In this work we study the structure of approximate solutions of variational problems with continuous integrands f: [0, ∞) × Rn × Rn → R1 which belong to a complete metric space of functions. The main result in this paper deals with the turnpike property of variational problems. To have this property means that the approximate solutions of the problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.

The main aim of this paper is to obtain optimality conditions for a constrained set-valued optimization problem. The concept of Clarke epiderivative is introduced and is used to derive necessary optimality conditions. In order to establish sufficient optimality criteria we introduce a new class of set-valued maps which extends the class of convex set-valued maps and is different from the class of invex set-valued maps.

In this paper we prove a generalization of the well known theorem of Krasnoselskii on the superposition operator in which the domain of Nemytskii's operator is a product space. We also give an application of this result.

In this paper we study the existence and uniqueness of a solution for minimization problems with generic increasing functions in an ordered Banach space X. The standard approaches are not suitable in such a setting. We propose a new type of perturbation adjusted for the problem under consideration, prove the existence and point out sufficient conditions providing the uniqueness of a solution. These results are proved by assuming that the space X enjoys the following property: each decreasing norm-bounded sequence has a limit. We supply a counterexample, which shows that this property is essential and give a modification of obtained results for the space C(T), which does not possess this property.