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\(\mathcal{H}_{2,\alpha}\)-norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations. (English) Zbl 1303.49014
Summary: This paper deals with a numerical solution method for optimal control problems subject to parabolic and hyperbolic evolution equations. Firstly, the problem is semi-discretized in space with the boundary or distributed controls as input and those parts of the discretized state appearing in the cost functional as output variables. The corresponding transfer function is then approximated optimally with respect to the \(\mathcal{H}_{2,\alpha}\)-norm providing an optimally reduced optimal control problem, which is finally solved by a first-discretize-then-optimize approach. To enable the application of this reduction method, a new constrained optimal model reduction problem subject to reduced systems with real system matrices is considered. Necessary optimality conditions and a transformation procedure for the reduced system to a canonical form of real matrices are presented. The method is illustrated with numerical examples where also complicated controls with many bang-bang arcs are investigated. The approximation quality of the optimal control and its correlation to the decay rate of the Hankel singular values of the system are studied numerically. A comparison to the approach of using balanced truncation for model reduction is applied.
MSC:
49M25 Discrete approximations in optimal control
49K20 Optimality conditions for problems involving partial differential equations
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
93C20 Control/observation systems governed by partial differential equations
Software:
BNDSCO; Ipopt
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