This module introduces the theory of optimal control, which involves control laws that maximise defined performance measures of linear and nonlinear dynamical systems. Optimal control problems are infinite-dimensional, and special emphasis is given to the use of computer tools for their solution. The topics covered include calculus of variations, Pontryagin's maximum principle, and dynamic programming. Part of the module will involve calculating numerically solutions to optimal control problems with the use of modern optimal control software.

By the end of this course you will be able to: 1 Formulate dynamic optimisation problems of practical significance, for linear and nonlinear systems, as optimal control problems. 2 Specify precise statements of the maximum principle for various general categories of optimal control problems, including free-time, fixed-time, free-endpoint, and fixed-endpoint problems. 3 Derive solutions to continuous-time optimal control problems by application of the maximum principle. 4 Signify solution concepts for optimal control problems, including bang-bang solutions and singular controls. 5 Calculate solutions to optimal control problems by dynamic programming techniques.

Calculus of variations: Euler-Lagrange equation, Hamiltonian formalism, variational problems with constraints. Pontryagin's maximum principle: proof of PMP, transversality condition, nonlinear two-point boundary-value problem, path constraints, corner conditions, bang-bang control, singular arcs. Dynamic programming: discrete problem, principle of optimality, Hamilton-Jacobi-Bellman equation, linear quadratic optimal control. Optimal control numerical solver software.