Bolza-type Problem Research Articles

A coupled problem described by time-dependent subdifferential operator and non-convex perturbed sweeping process

The aim of the present paper is to state the well-posedness of a new dynamical system driven by a differential inclusion involving time-dependent subdifferential operators with integral perturbation and a non-convex perturbed sweeping process. The current study is motivated by recent works on integro-differential sweeping processes and dynamical systems coupled by differential inclusions (involving sweeping processes and maximal monotone operators). In our development, we proceed using a discretization method, combining tools of first-order evolution problems of subdifferential type and those governed by the normal cone of $ r $-prox regular moving sets. We establish a new existence result to the class of dynamical systems under consideration, in the context of infinite dimensional Hilbert setting. This allows us to deal with a Bolza-type problem in optimal control theory.

On the Accuracy of the Model Predictive Control Method

The paper investigates the accuracy of the model predictive control (MPC) method for finding on-line approximate optimal feedback control for Bolza-type problems on a fixed finite horizon. The predictions for the dynamics, the state measurements, and the solution of the auxiliary open-loop control problems that appear at every step of the MPC method may be inaccurate. The main result provides an error estimate of the MPC-generated solution compared with the optimal open-loop solution of the “ideal” problem, where all predictions and measurements are exact. The technique of proving the estimate involves an extension of the notion of strong metric subregularity of set-valued maps and utilization of a specific new metric in the control space, which makes the proof nonstandard. The result is specialized for two problem classes: coercive problems and affine problems.

A control problem involving time and state-dependent maximal monotone operators via Young measures

The article deals with a minimizing problem related to a differential inclusion governed by time- and state-dependent maximal monotone operators and single-valued perturbations with Young measures. A Bolza-type problem and relaxation are studied.

Optimal Control Problems Governed by Time Dependent Subdifferential Operators with Delay

ABSTRACTThis article is devoted to the study, in the infinite dimensional setting, of a Bolza-type problem governed by a class of functional evolution inclusion which involves a time-dependent subdifferential operator with a time-delay perturbation. We present a relaxation result associated with such equations where the controls are Young measures in order to show the existence of an optimal solution under a suitable convexity assumption.

A control problem governed by a second-order differential inclusion

This article studies some Bolza-type problems governed by second-order differential inclusions with two boundary conditions, where the controls are Young measures.

Epi-convergent discretization of the generalized Bolza problem in dynamic optimization

The paper is devoted to well-posed discrete approximations of the so-called generalized Bolza problem of minimizing variational functionals defined via extended-real-valued functions. This problem covers more conventional Bolza-type problems in the calculus of variations and optimal control of differential inclusions as well of parameterized differential equations. Our main goal is find efficient conditions ensuring an appropriate epi-convergence of discrete approximations, which plays a significant role in both the qualitative theory and numerical algorithms of optimization and optimal control. The paper seems to be the first attempt to study epi-convergent discretizations of the generalized Bolza problem; it establishes several rather general results in this direction.

A variational principle for gradient flows

We verify – after appropriate modifications – an old conjecture of Brezis-Ekeland ([3], [4]) concerning the feasibility of a global variational approach to the problems of existence and uniqueness of gradient flows for convex energy functionals. Our approach is based on a concept of ‘‘self-duality’’ inherent in many parabolic evolution equations, and motivated by Bolza-type problems in the classical calculus of variations. The modified principle allows to identify the extremal value –which was the missing ingredient in [3]– and so it can now be used to give variational proofs for the existence and uniqueness of solutions for the heat equation (of course) but also for quasi-linear parabolic equations, porous media, fast diffusion and more general dissipative evolution equations.

An existence theorem for neutral variational problems of Bolza

This note proves an existence theorem for a generalized Bolza-type problem that has time delays in both the state and velocity variables. The assumptions are stated in terms of a modification of the classical Hamiltonian, and extend ideas of Rockafellar to the delay case.

A quadratic Bolza-type problem in a Riemannian manifold

A classical nonlinear equation D s x ̇ + ∇ xV(x,s)=0 on a complete Riemannian manifold M is considered. The existence of solutions connecting any two points x 0,x 1∈ M is studied, i.e., for T>0 the critical points of the functional J T(x)= 1 2 ∫ 0 T 〈 x ̇ , x ̇ 〉 ds− ∫ 0 T V(x,s)ds with x(0)= x 0, x( T)= x 1. When the potential V has a subquadratic growth with respect to x, J T admits a minimum critical point for any T>0 (infinitely many critical points if the topology of M is not trivial). When V has an at most quadratic growth, i.e., V(x,s)⩽λd 2(x, x ̄ )+k , this property does not hold, but an optimal arrival time T( λ)>0 exists such that, if 0< T< T( λ), any pair of points in M can be joined by a critical point of the corresponding functional. For the existence and multiplicity results, variational methods and Ljusternik–Schnirelman theory are used. The optimal value T(λ)=π/ 2λ is fulfilled by the harmonic oscillator. These ideas work for other related problems.

Analysis of generalized guidance laws for homing missiles

It is shown that the optimal guidance problem can be handled as a Bolza-type problem which can be solved by straightforward calculation. The solution proves to be different from that given by Yang, et al. (1989) and is verified to be a minimum performance index solution. Also presented is an alternate unified approach to the solution of the nonlinear system of differential equation of the generalized proportional navigation law discussed earlier by Yang, et al. This unified equation has been solved for all the different cases of proportional navigation for which closed-form solutions have been published by earlier authors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>