# Necessary Optimality Conditions In Discrete Nonsmooth Optimal Control

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### www.esaim-cocv.org

ESAIM: COCV ESAIM: Control, Optimisation and Calculus of Variations April 2005, Vol. 11, 285 309 DOI: 10.1051/cocv:2005008 OPTIMAL CONTROL OF DELAY SYSTEMS WITH DIFFERENTIAL AND

### International Journal Of Mathematical Analysis And

This book is a self-contained elementary study for nonsmooth analysis and optimization, and their use in solution of nonsmooth optimal control problems. The first part of the book is concerned with nonsmooth differential calculus containing necessary tools for nonsmooth optimization.

### Discrete Maximum Principle for Nonsmooth Optimal Control

Abstract. We consider optimal control problems for discrete-time systems with delays. The main goal is to derive necessary optimality conditions of the discrete maximum principle typf' in the case of nonsmooth minimizing functions. We obtain two independent forms of the di~

### Identification of some nonsmooth evolution systems with

leading both to sharp necessary optimality conditions as well as to an efﬁcient numerical procedure based on the so-called Implicit Programming approach (ImP), cf. [4,5]. In particular, on the basis of the subdifferential calculus of Mordukhovich [6,7] we will show thatthesolutionmap S: π → (u,z

### Optimality Conditions by Means of the Generalized HJB Equation

Necessary and sufficient conditions of optimality for partial information control problems are given in 2 In [11], the sufficient maximum principle and its link with the dynamic programming principle are discussed. The second order stochastic maximum principle for optimal controls of nonlinear

### Research Article Optimal Control Problem for Switched

yield the necessary optimality conditions (i) (v) in eo-rem is completes the proof of the theorem. eorem (Necessary optimality conditions for switching Let the minimization functional 9 (A) be positively homogenous, quasidi erentiable at a point (A),andlet(/ (A), (A),C) be an optimal solution to the control problem ( ) ( ).en,

### ON OPTIMAL CONTROL OF A SWEEPING PROCESS COUPLED

cretization and the coderivative calculus to derive necessary optimality conditions for the original problem. However, their control enters the system via the sweeping set and so the results of [10] cannot be compared with those presented in this paper in a straightforward way. The aim of this paper is

### A List of Publications of Prof. Yang Xiaoqi (January 2019)

A nonsmooth version of alternative theorem and nons-mooth multiobjective programming, Utilitas Mathematica optimal control problem, 1996 Special Volume of the Far East J. Math. Sci and Teo, K.L., Necessary optimality conditions for bicriteria discrete op-timal control problems, Journal of the Australian Mathematical Society

### The Approximate Maxium Principle in Constrained Optimal

Abstract. The paper concerns optimal control problems for dynamic systems governed by a parametric family of discrete approximations of control systems with continuous time. Discrete approximations play an important role in both qualitative and numerical aspects of optimal control and occupy an intermediate

### Optimal Control Problems For Partial Differential

This book focuses on optimal control problems where the state equation is an elliptic or parabolic partial differential equation. Included are topics such as the existence of optimal solutions, necessary optimality conditions and adjoint equations, second-order sufficient conditions, and main principles of selected numerical techniques.

### Jane (Juan-Juan) Ye - UVic.ca

71. L. Guo* and J.J. Ye, Necessary optimality conditions for optimal control problems with equilibrium constraints, SIAM Journal on Control and Optimization, 54 (2016), 2710-2733. 70. A. Li and J.J. Ye, Necessary optimality conditions for optimal control problems with nonsmooth mixed state and control constraint, Set-valued and Variational

### Variational Analysis in Nonsmooth Optimization and

Variational Analysis in Nonsmooth Optimization and Discrete Optimal Control Boris S. Mordukhovich Department of Mathematics, Wayne State University, Detroit, Michigan 48202, [email protected] This paper is devoted to applications of modern methods of variational analysis to constrained optimization and control

### Polyhedral optimization of second-order discrete and

hedral constraints. Due to the use of convex and nonsmooth analysis structures, the necessary and suﬀicient conditions of optimality are derived for this problem in terms of the Euler Lagrange polyhedral inclusions. Note that the reasons for adopting discrete modeling are as follows: First, statistics are collected at discrete times

### Necessary Optimality Condition with Feedback Controls for

Necessary Optimality Condition with Feedback Controls for Nonsmooth Optimal Impulsive Control Problems Stepan P. Sorokin Matrosov Institute for System Dynamics and Control Theory SB RAS Lermontova St., 134, 664033 Irkutsk, Russia [email protected] Maxim V. Staritsyn Matrosov Institute for System Dynamics and Control Theory SB RAS Lermontova St., 134,

### Universit at Regensburg Mathematik

2016] we derive necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. eW present numerical exam-ples with distributed and boundary controls, and also consider the case, where the initial aluev of the phase eld serves as control ariable.v Keywords: Optimal control, Boundary control, Initial aluev

### Optimal Control, Stabilization and Nonsmooth Analysis

Necessary and SufHcient Conditions for Turnpike Optimality: The Singular Case Alain Rapaport, Pierre Cartigny 85 Min Plus Eigenvector Methods for Nonlinear Hoc Problems with Active Control Gaemus Collins, William McEneaney 101 Optimization and Feedback Control of Constrained Parabolic Systems under Uncertain Perturbations

### Approximate Maximum Principle for Discrete Approximations

Approximate Maximum Principle for Discrete Approximations of Optimal Control Systems with Nonsmooth Objectives and Endpoint Constraints BORIS S. MORDUKHOVICH1 Department of Mathematics Wayne State University Detroit, MI 48202 Email: [email protected] ILYA SHVARTSMAN Department of Mathematics and Computer Science Penn State University - Harrisburg

### 1 Introduction - Optimization Online

control. In this way we establish necessary optimality conditions for optimal solutions to di erential inclusions and discuss their various applications. Deriving necessary optimality conditions strongly involves advanced tools of rst-order and second-order variational analysis and generalized di erentiation. Key words. Optimal control, di erential inclusions, variational analysis, sweeping processes, discrete

### Discrete Time Pontryagin Principles with Infinite Horizon

We establish necessary conditions of optimality for problems of optimal control theory in the discrete time framework with infinite horizon. Our necessary condi-tions are in the form of Pontryagin principles. We treat smooth and partially nonsmooth settings, without

### Nonlinear Analysis and Optimization

the Pontryagin Maximum Principle for control problems with state and mixed con-straints, deep investigations of abnormal extremals, new controllability conditions to avoid the degeneracy phenomenon, second-order optimality conditions for singular and nonsingular extremals, necessary optimality conditions for Lipschitzian differ-B Boris S

### MESH-INDEPENDENCE OF SEMISMOOTH NEWTON METHODS

A class of nonlinear elliptic optimal control problems with mixed control-state con- straints arising, e.g., in Lavrentiev-type regularized state constrained optimal control is considered. Based on its ﬁrst order necessary optimality conditions, a semismooth Newton method is proposed

### Optimal control of hybrid systems in manufacturing

necessary conditions for optimality, introduce the concept of critical jobs, and identify several structural properties that allow for the decomposition of the overall nonconvex and nonsmooth problem into smaller and simpler ones. In Section III-B, we present an explicit algorithmic procedure

### Control and Cybernetics vol. 34 (2005) No. 3

variational analysis needed for deriving necessary optimality conditions in dis-crete approximation problems and then establishing, by passing to the limit, adequate necessary conditions for optimality of the given solution to (P). Section 4 is devoted to necessary optimality conditions for the discrete-time

### Nonsmooth Analysis in Systems and Control Theory

deriving or expressing necessary conditions, in applying suﬃcient conditions, or in studying the sensitivity of the problem. The need to consider nonsmoothness in the case of stabilizing (as opposed to optimal) control has come to light more recently. It appears in particular that in the analysis of truly nonlinear control systems, the consideration of nonsmooth Lyapunov functions and discontinuous

### Discrete Approximations of a Controlled Sweeping Process

ing control problem (P), the method of discrete approximations can be viewed as a driving force to derive necessary optimality conditions for continuous-time control prob-lems. In the framework of (P), this requires deriving necessary optimality conditions for the (nonsmooth) discrete approximation problems and then passing there to the limit

### Boris S. Mordukhovich and Ilya Shvartsman

ﬁrst result in the literature that holds for nonsmooth objectives and endpoint constraints. The results obtained establish necessary optimality conditions for constrained nonconvex ﬁnite-diﬀerence control systems and justify stability of the Pontryagin Maximum Principle for continuous-time systems under discrete approximations.

### Technische Universitat M unchen

the proposed nonsmooth and nonconvex optimal control problem. A proof of existence of minimizers is given and a rigorous formulation and derivation of rst-order optimality conditions of the optimal control problem are provided. The main contribution on the numerical side is the analysis of a time stepping

### Necessary conditions of optimality for measure driven

The stated necessary conditions optimality are in the form of both an Hamiltonian inclusion and a maximum principle providing a complete characterization of both optimal and singular - notably that during jumps - optimal evolutions of the state trajectory.

### THE EXTENDED EULER{LAGRANGE CONDITION

This paper provides necessary conditions of optimality for a general variational formulation traditionally adopted in optimal control theory involving a di erential equation parameterized by a control function u x (t)=f namely conditions which are nonsmooth analogues of

### On the Relation between the Minimum Principle and Dynamic

the MP and DP constitute necessary conditions for optimality which under certain assumptions become sufcient (see e.g. [4] [8]). However, Dynamic Programming is widely used as a set of sufcient conditions for optimality after the optimal control extension of Carath eodory's sufcient conditions in´ Calculus of Variations (see e.g. [4] [6]).

### EULER-LAGRANGE AND HAMILTONIAN FORMALISMS IN

of necessary optimality conditions were obtained that either followed the patterns of the classical Euler{Lagrange or Hamiltonian formalisms or were based on the Pontriagin maximum principle for parametrized problems of optimal control. In each case two fundamental relations associated with the di erential inclusion

### First order optimality conditions under state constraints

Many optimal control problems represent the real life behaviour of technical, biological or economical systems. Thus the limited physical or economical resources necessitate the use of constraints imposed on state or control variables. Necessary conditions of optimality must be formulated to

### LAGRANGIAN AND HAMILTONIAN NECESSARY CONDITIONS

In [7], Ioﬀe used the result established in [8] to derive necessary optimality conditions for optimal control problems of unbounded integrably sub-Lipschitz (in the sense of Loewen-Rockafellar) diﬀerential inclusions. Based on the result by Ioﬀe-Rockafellar, Vinter and Zheng [25] provide necessary optimality conditions for problems involv-

### A convex analysis approach to optimal controls with

Abstract Optimal control problems involving hybrid binary continuous control costs are partial (Moreau Yosida) regularization of nonsmooth convex ˙nite-dimensional problems for [20, Proposition 4.4.4] for @G to arrive at the necessary optimality conditions (2.1).

### Optimal Control Problems For Partial Differential

hand sides, the existence of optimal controls, the necessary conditions of optimality, the controllability of systems, numerical methods of approximation of generalized solutions of initial boundary value problems with generalized data, and numerical methods for approximation of optimal controls.

### Institute of Applied Mechanics, Technical University of

the optimality conditions and for the numerical treatment certain results from the theory of optimal control of variational inequalities are applied (see e.g. [5], [6], [7]). Two-level solution algorithms, where the structural analysis problem is solved in the lower level and the optimal control pro-

### Conjugate Duality and the Control of Linear Discrete Systems

necessary and su cient optimality conditions and study the existence of optimal solutions for the primal. In Section 4 we apply the general results in the context of the control of linear discrete systems and rediscover some results from the literature as particular cases. 2 Elements of Convex Analysis

### Monday, September 11th COMPARISON OF NONDEGENERATE

ROBUST DISCRETE TIME SIMPLIFIED ADAPTIVE CONTROL Rabin Ben Yamin, Isaac Yaesh, Uri Shaked CONTROL OF THE HIMAT AIRCRAFT VIA ALGEBRAIC µ-SYNTHESIS Marek Dlapa, Roman Prokop MA-7 11:30 AM-12:30 PM 02.3 FUZZY CONTROL APPLICATIONS ROBUST PREDICTIVE FUZZY CONTROL Jana Paulusová, Stefan Kozák NAVIGATION OF A MOBILE ROBOT USING FUZZY

### INTERNATIONAL CONFERENCE on CONTROL THEORY AND ITS

Necessary and Sufficient Conditions for Weak Local Minima in Optimal Control E. Khmelnitsky & K. Kogan Necessary Optimality Conditions for a Generalized Problem of Production Scheduling M. Drakhlin & E. Litsyn On the Variational Problem in the Space of When can Kalman's Optimal Discrete State Control-Law be Realized Via Output Feedback

### DIDOManualPR1

3 Solution satisfy necessary optimality conditions Cons 3 Necessary optimality conditions must be derived analytically (most likely hard!) 3 Small radius of convergence: Need a good initial guess 3 Need a guess for costates 3 Need to know a priori the constrained and unconstrained sub arcs Direct Methods Transform the continuous OCP into a discrete NLP

### JOURNAL OF LA A Maximum Principle for Hybrid Optimal

set of ﬁrst order necessary conditions of optimality, akin t o the traditional maximum principle, relating to a selection of continuous variables in a hybrid optimal control problem, that optimizes the cost function for a ﬁxed choice of the discrete variables. For some simple cases of hybrid optimal

### Chance-Constrained Sequential Convex Programming for

guaranteed to satisfy ﬁrst-order necessary optimality conditions and all constraints, which is critically en-abled by our discrete-time C1 problem formulation. This paper is organized as follows. In Section II, we state the chance-constrained optimal control problem. In Section III, we review known approaches to model uncertainty and

### Optimal Control of Partial Differential Equations with

3 Multi-bang control Multi-bang control refers to optimal control problems for par-tial differential equations where a distributed control should only take on values from a discrete set of values u i. This prop-erty can be promoted by a combination of L2 and L0-type con-trol costs. The resulting functional, however, is non-convex

### A convex analysis approach to optimal controls with

Optimal control problems involving hybrid binary continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and

### Miami University October 12{13, 2018

Optimal Control of Sweeping Processes with Applications (Boris Mor- 12 Oct Cone and Nonsmooth Calculus 10:00-10:25 am BAC 102 Douglas Ward Miami University but not necessary for the second-order growth condition. It can be used to characterize strong local minimiz-

### The maximum principle for discrete delay inclusions with

This idea has been, already used in [6], [3] and [5] to obtain necessary optimality conditions for optimal control problems given by differential inclusions, hyper-bolic differential inclusions and discrete inclusions. Finally, a last step uses the Minchenko and Sirotko duality results in

### METHOD OF DISCRETE APPROXIMATIONS FOR CONTROLLED

for deriving necessary optimality conditions in optimal control of the di↵erential inclusions x˙(t) 2 F ⇣ t,x(t) ⌘ a.e. t 2 [0,T] was developed in [Mor95,06] for bounded Lipschitzian map-pings F(t, ). The three major steps of this approach are: Replace the time derivative x˙(t) by the ﬁnite di↵erences x˙(t) ⇡

### Optimal control of a class of hybrid systems - Automatic

necessary conditions for optimality, introduces the nonsmooth optimization elements needed to handle the nondifferentiabili-ties involved, and concludes with a theorem that characterizes an optimal control sequence. Section IV presents several prop-erties of the optimal solutions and introduces the concept of