Nikolai P. Osmolovskii

Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control

This book is devoted to the theory and applications of second-order necessary and sufficient optimality conditions in the calculus of variations and optimal control. The authors develop theory for a control problem with ordinary differential equations subject to boundary conditions of equality and inequality type and for mixed state-control constraints of equality type. The book is distinctive in that necessary and sufficient conditions are given in the form of no-gap conditions; the theory covers broken extremals where the control has finitely many points of discontinuity; and a number of numerical examples in various application areas are fully solved. Audience: This book is suitable for researchers in calculus of variations and optimal control and researchers and engineers in optimal control applications in mechanics; mechatronics; physics; economics; and chemical, electrical, and biological engineering. Contents: List of Figures; Notation; Preface; Introduction; Part I: Second-Order Optimality Conditions for Broken Extremals in the Calculus of Variations; Chapter 1: Abstract Scheme for Obtaining Higher-Order Conditions in Smooth Extremal Problems with Constraints; Chapter 2: Quadratic Conditions in the General Problem of the Calculus of Variations; Chapter 3: Quadratic Conditions for Optimal Control Problems with Mixed Control-State Constraints; Chapter 4: Jacobi-Type Conditions and Riccati Equation for Broken Extremals; Part II: Second-Order Optimality Conditions in Optimal Bang-Bang Control Problems; Chapter 5: Second-Order Optimality Conditions in Optimal Control Problems Linear in a Part of Controls; Chapter 6: Second-Order Optimality Conditions for Bang-Bang Control; Chapter 7: Bang-Bang Control Problem and Its Induced Optimization Problem; Chapter 8: Numerical Methods for Solving the Induced Optimization Problem and Applications; Bibliography; Index.

Necessary Conditions for a Weak Minimum in Optimal Control Problems with Integral Equations Subject to State and Mixed Constraints

The first order necessary optimality conditions for a weak minimum are derived for optimal control problems with Volterra-type integral equations, considered on a fixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the uniform linear-positive independence of the gradients of active mixed constraints with respect to the control. The conditions obtained generalize the Euler--Lagrange equation (as a stationarity condition) for the Lagrange problem in the classical calculus of variations with ordinary differential equations.

Second-Order Necessary Optimality Conditions for the Mayer Problem Subject to a General Control Constraint

This paper is devoted to second-order necessary optimality conditions for the Mayer optimal control problem with an arbitrary closed control set \(U \subset \mathbb{R}^{m}\). Admissible controls are supposed to be measurable and essentially bounded. Using second order tangents to U, we first show that if \(\bar{u}(\cdot )\) is an optimal control, then an associated quadratic functional should be nonnegative for all elements in the second order jets to U along \(\bar{u}(\cdot )\). Then we specify the obtained results in the case when U is given by a finite number of C2-smooth inequalities with positively independent gradients of active constraints. The novelty of our approach is due, on one hand, to the arbitrariness of U. On the other hand, the proofs we propose are quite straightforward and do not use embedding of the problem into a class of infinite dimensional mathematical programming type problems. As an application we derive new second-order necessary conditions for a free end-time optimal control problem in the case when an optimal control is piecewise Lipschitz.

Second-order conditions for optimal control problems with mixed control-state constraints and control appearing linearly

We study optimal control problems with a two-sided mixed control-state constraint and assume that the control variable appears linearly in both the system dynamics and constraints. By defining the control-state constraint as a new control variable, the optimal control problem is transformed into an optimal control problem with simple bounds on the new control variable. In view of Pontryagins Minimum Principle, optimal controls of the transformed problem are concatenations of bang-bang or singular arcs. Second-order sufficient conditions (SSC) for such bang-singular controls have recently been given in the literature. We summarize results on SSC and illustrate their numerical verification on the optimal control of the Rayleigh equation for various bounds in a two-sided control-state constraint.

On the Proof of Pontryagin’s Maximum Principle by Means of Needle Variations

We propose a proof of the maximum principle for the general Pontryagin type optimal control problem, based on packets of needle variations. The optimal control problem is first reduced to a family of smooth finite-dimensional problems, the arguments of which are the widths of the needles in each packet, then, for each of these problems, the standard Lagrange multipliers rule is applied, and finally, the obtained family of necessary conditions is “compressed” in one universal optimality condition by using the concept of centered family of compacta.

Strong Local Minimizers in Optimal Control Problems with State Constraints: Second-Order Necessary Conditions

This paper is devoted to second-order necessary optimality conditions for strong local minima for a Mayer type optimal control problem with a general control constraint

A general lagrange multipliers theorem

U \subset {\mathbb R}^m

Second-Order Optimality Conditions for Broken Extremals and Bang-Bang Controls: Theory and Applications

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Necessary optimality conditions in a problem with integral equations on a nonfixed time interval subject to mixed and state constraints

We consider a general optimization problem with equality and inequality constraints in a Banach space, the latter being given by closed convex cones with nonempty interiors. A necessary optimality condition in the form of Lagrange multipliers rule is presented, that is convenient for application to a wide range of optimization problems.

Second Order Sufficient Conditions for Time-Optimal Bang-Bang Control

We survey the results on no-gap second-order optimality conditions (both necessary and sufficient) in the Calculus of Variations and Optimal Control, that were obtained in the monographs Milyutin and Osmolovskii (Calculus of Variations and Optimal Control. Translations of Mathematical Monographs. American Mathematical Society, Providence, 1998) and Osmolovskii and Maurer (Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control. SIAM Series Design and Control, vol. DC 24. SIAM Publications, Philadelphia, 2012), and discuss their further development. First, we formulate such conditions for broken extremals in the simplest problem of the Calculus of Variations and then, we consider them for discontinuous controls in optimal control problems with endpoint and mixed state-control constraints, considered on a variable time interval. Further, we discuss such conditions for bang-bang controls in optimal control problems, where the control appears linearly in the Pontryagin-Hamilton function with control constraints given in the form of a convex polyhedron. Bang-bang controls induce an optimization problem with respect to the switching times of the control, the so-called Induced Optimization Problem. We show that second-order sufficient condition for the Induced Optimization Problem together with the so-called strict bang-bang property ensures second-order sufficient conditions for the bang-bang control problem. Finally, we discuss optimal control problems with mixed control-state constraints and control appearing linearly. Taking the mixed constraint as a new control variable we convert such problems to bang-bang control problems. The numerical verification of second-order conditions is illustrated on three examples.