See also Becerra, for a straightforward way of combining a dynamic simulation tool with nonlinear programming code to solve optimal control problems with constraints. Indirect methods involve iterating on the necessary optimality conditions to seek their satisfaction.

This usually involves attempting to solve nonlinear two-point boundary value problems, through the forward integration of the plant equations and the backward integration of the co-state equations. Examples of indirect methods include the gradient method and the multiple shooting method, both of which are described in detail in the book by Bryson Dynamic programming is an alternative to the variational approach to optimal control. It was proposed by Bellman in the s, and is an extension of Hamilton-Jacobi theory.

Bellman's principle of optimality is stated as follows: "An optimal policy has the property that regardless of what the previous decisions have been, the remaining decisions must be optimal with regard to the state resulting from those previous decisions". This principle serves to limit the number of potentially optimal control strategies that must be investigated. It also shows that the optimal strategy must be determined by working backward from the final time.

Consider Problem 1 with the addition of a terminal state constraint In some cases, the HJB equation can be used to find analytical solutions to optimal control problems.

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Dynamic programming includes formulations for discrete time systems as well as combinatorial systems, which are discrete systems with quantized states and controls. Discrete dynamic programming, however, suffers from the 'curse of dimensionality', which causes the computations and memory requirements to grow dramatically with the problem size.

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See the books Lewis and Syrmos, , Kirk, , and Bryson and Ho, for further details on dynamic programming. Most of the problems defined above have discrete-time counterparts. These formulations are useful when the dynamics are discrete for example, a multistage system , or when dealing with computer controlled systems. In discrete-time, the dynamics can be expressed as a difference equation:.

This example is solved using a gradient method in Bryson, Here, a path constraint is considered and the solution is sought by using a direct method and nonlinear programming.

## Control and Optimal Control Theories with Applications

Such a flight path may be of interest to reduce engine noise over populated areas located ahead of an airport runway. This manoeuvre can be formulated as an optimal control problem, as follows. The distance and time units in the above equations are normalised. To obtain meters and seconds, the corresponding variables need to be multiplied by The solution shown in Fig 2 was obtained by using sequential quadratic programming, where the decision vector consisted of the control values at the grid points.

### Description

Becerra , Scholarpedia, 3 1 Jump to: navigation , search. Post-publication activity Curator: Victor M. Becerra Contributors:. Subbaram Naidu. Sponsored by: Eugene M. Izhikevich , Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia Reviewed by : Dr.

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Categories : Control Theory Dynamical Systems. Namespaces Page Discussion. Thomas Weber. Content This book bridges optimal control theory and economics, discussing ordinary differential equations, optimal control, game theory, and mechanism design in one volume. Thomas A. Weber Foreword by A. Author: Alexandra von Schack. Source: Management of Technology and Entrepreneurship.

Existence Theorems for Optimal Control Problems. Back Matter Pages About this book Introduction This monograph is an introduction to optimal control theory for systems governed by vector ordinary differential equations.

## Optimal Control Blog

It is not intended as a state-of-the-art handbook for researchers. We have tried to keep two types of reader in mind: 1 mathematicians, graduate students, and advanced undergraduates in mathematics who want a concise introduction to a field which contains nontrivial interesting applications of mathematics for example, weak convergence, convexity, and the theory of ordinary differential equations ; 2 economists, applied scientists, and engineers who want to understand some of the mathematical foundations.

In general, we have emphasized motivation and explanation, avoiding the "definition-axiom-theorem-proof" approach.