Analysis and Computation for Optimal Control Problems with Pointwise State and Control Constraints

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9803755DunnPhysical systems whose states evolve in response to variable externallyapplied controlling forces, voltages, temperatures, interest rate policies,resource allocation strategies, and the like, are commonly modeled bydifferential equations or approximating difference equations. Relatedoptimization problems arise naturally when some choice is permitted inthe way the control inputs are applied in time or space. These optimal control problems have classic precursors in the Calculus of Variations, and may also be viewed as specially structured nonlinear programs in function spaces orfinite-dimensional sequence spaces. The most difficult optimal controlproblems enforce strict pointwise bounds or other inequality constraintson the control and state variables. Run-of-the-mill problems in thiscategory can easily entail hundreds or thousands of variables with comparablymany constraints, and when cast in the currently accepted mathematical format,are often so badly scaled that the simplest and most readily implementediterative optimization methods are effectively incapacitated. The moresophisticated Newtonian and quasi-Newtonian algorithms do providecomputational countermeasures for bad scaling near almost-singular local minimizers; however, Newtonian scaling procedures are costly and do notguarantee good nonlocal convergence properties while the iterates are stillfar from a local minimizer.The proposed investigation would address these issues with theoretical andcomputational evaluations of standard optimization schemes implemented ina new nonstandard mathematical framework for optimal control problemswith pointwise state and control constraints. The alternative framework replaces local differential or difference forms of the state evolution equations by nonlocal integrated forms, and replaces state variables in the primal variable set by new artificial variables that coincide with the state when all constraints are met. Recent experiments with hybrid augmented Lagrangian projection methods outlined in this proposal indicate real and important analytical and computational advantages in the new formulation. The goal of the proposed study is to gain more experience with algorithm implementations in this setting, and to develop sharpened optimality conditions and other mathematical tools needed to achieve a deeper understanding of the observed behavior and improve the computational methods. Improvements in this area must have an immediate and significant practical impact, since large-scale computationally challenging optimal control problems arise in many physical settings.