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In this study, first-order necessary optimality conditions, in the form of a weak maximum principle, are derived for discrete optimal control problems with mixed equality and inequality constraints. Such conditions are achieved by using the Dubovitskii–Milyutin formalism approach. Nondegenerate conditions are obtained under the constant rank of the subspace component (CRSC) constraint qualification, which is an important generalization of both the Mangasarian–Fromovitz and constant rank constraint qualifications. Beyond its theoretical significance, CRSC has practical importance because it is closely related to the formulation of optimization algorithms. In addition, an instance of a discrete optimal control problem is presented in which CRSC holds while other stronger regularity conditions do not. [ABSTRACT FROM AUTHOR]

We study the approximate controllability of the discrete fractional systems of order 1 < α < 2 (∗) C ∇ α u n = A u n + B v n + f (n , u n) , n ≥ 2 , subject to the initial states u 0 = x 0 , u 1 = x 1 , where A is a closed linear operator defined in a Hilbert space X, B is a bounded linear operator from a Hilbert space U into X , f : N 0 × X → X is a given sequence and C ∇ α u n is an approximation of the Caputo fractional derivative ∂ t α of u at t n : = τ n , where τ > 0 is a given step size. To do this, we first study resolvent sequences { S α , β n } n ∈ N 0 generated by closed linear operators to obtain some subordination results. In addition, we discuss the existence of solutions to (∗) and next, we study the existence of optimal controls to obtain the approximate controllability of the discrete fractional system (∗) in terms of the resolvent sequence { S α , β n } n ∈ N 0 for some α , β > 0. Finally, we provide an example to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper studies the solution stability of convex optimization and discrete convex optimal control problems in Banach spaces, where the solution set may be empty. For both the optimization problem and the optimal control problem, formulas for the ε -subdifferential of the optimal value function are derived without qualification conditions. We first calculate the ε -subdifferential of the optimal value function to a parametric optimization problem with geometrical and functional constraints. We then use the obtained results to derive a formula for computing the ε -subdifferential of the optimal value function to a discrete convex optimal control problem with linear state equations, control constraints and initial, terminal conditions. [ABSTRACT FROM AUTHOR]

In this paper, we derive formulae for computing the S-derivative of the extremum multifunction in a multi-objective parametric discrete optimal control problem with nonconvex objective functions and control constraints. Particularly, we obtain formulae for upper and lower evaluation on the S-derivative of the extremum multifunction via the solution of state equations, the tangent cone to the constraint sets, and the Fréchet derivative of the objective functions. By establishing an abstract result on the S-derivative of the extremum multifunction in a multi-objective parametric mathematical programming problem, we derive formulae for upper and lower evaluation on the S-derivative of the extremum multifunction in a multi-objective parametric discrete optimal control problem. [ABSTRACT FROM AUTHOR]

This work concerns with semi-Markov decision chains evolving on a finite state space. The controller has a positive and constant risk sensitivity coefficient, and the performance of a control policy is measured by the risk-sensitive average cost criterion. Under conditions ensuring that the optimal value function is determined via a single optimality equation, the fixed points of a family of contractive operators are used to obtain convergent approximations to the optimal average cost and to a solution of optimality equation, extending the classical discounted approach to the context of the paper. In contrast with the Markovian case, the contractive operators utilized in this work depend on two parameters. [ABSTRACT FROM AUTHOR]

This work concerns with discrete-time Markov decision processes on a denumerable state space. Assuming that the decision maker is risk-averse with constant risk-sensitivity coefficient, the performance of a control policy is measured by an average criterion associated with a non-negative and bounded cost function. Under conditions ensuring that the optimal average cost is constant, but not necessarily determined via the average cost optimality equation, it is shown that a discounted criterion can be used to approximate the optimal average index. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem governed by a system of nonlinear difference equations. We obtain the existence of the optimal control as well as first-order optimality conditions of Pontryagin type by using the Dubovitskii–Milyutin formalism. Also, we give the necessary and sufficient conditions for global optimality. [ABSTRACT FROM AUTHOR]

In this paper, first-order and second-order necessary conditions for optimality for discrete-time stochastic optimal control problems governed by discrete-time Itô equations are established. A new discrete-time backward stochastic equation and discrete-time backward stochastic matrix equation are introduced. Based on the discrete-time backward stochastic Itô equation, a discrete-time stochastic maximum principle for the stochastic discrete optimal control problems is obtained. Moreover, using the discrete-time backward stochastic matrix equation the second-order necessary conditions for optimality are obtained. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the determination of periodic state trajectories of discrete-time dynamic systems resulting from the application of bidirectional on/off input sequences subject to switching constraints. The main contribution consists of characterizing the feasible input sequences and associated state trajectories in terms of a set of linear constraints involving integer variables. For practical purposes, the resulting solutions obtained by an integer linear solver can be evaluated in terms of engineering criteria such as amplitude or fuel expense in order to choose an appropriate trajectory to be used as target in a control law. A numerical example involving a double-integrator system is presented to illustrate the use of the proposed constraint formulation for the determination of feasible periodic state trajectories. [ABSTRACT FROM AUTHOR]

Motivated by our recent works on optimality conditions in discrete optimal control problems under a nonconvex cost function, in this paper, we study second-order necessary and sufficient optimality conditions for a discrete optimal control problem with a nonconvex cost function and state-control constraints. By establishing an abstract result on second-order optimality conditions for a mathematical programming problem, we derive second-order necessary and sufficient optimality conditions for a discrete optimal control problem. Using a common critical cone for both the second-order necessary and sufficient optimality conditions, we obtain 'no-gap' between second-order optimality conditions. [ABSTRACT FROM AUTHOR]

This paper provides a characterization of the optimal average cost function, when the long-run (risk-sensitive) average cost criterion is used. The Markov control model has a denumerable state space with finite set of actions, and the characterization presented is given in terms of a system of local Poisson equations, which gives as a by-product the existence of an optimal stationary policy. [ABSTRACT FROM AUTHOR]

In the paper, a potential-based policy iteration method is proposed for optimal control of a stochastic dynamic system with an average cost criterion and a parameterized control law. In this method, the potential function and the optimal control parameters are obtained via a least-squares-based approach. The potential estimation algorithm is derived from a temporal difference learning method, which can be viewed as a continuous version of the least-squares policy evaluation algorithm. The policy iteration algorithm is validated by solving a linear quadratic gaussian problem in the simulation. [ABSTRACT FROM AUTHOR]

This note provides a method to estimate the infinite-time performance of nonlinear model predictive control schemes. Based on principle of optimality, the upper and lower bounds of the ratio between the costs of model predictive control and finite-horizon optimal control are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study second-order necessary optimality conditions for a discrete optimal control problem with a nonconvex cost function and control constraints. By establishing an abstract result on second-order necessary optimality conditions for a mathematical programming problem, we derive second-order optimality conditions for a discrete optimal control problem. [ABSTRACT FROM AUTHOR]