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POD-Galerkin model order reduction for parametrized time dependent linear quadratic optimal control problems in saddle point formulation. (English) Zbl 1444.49013
In a parametrized optimal control problem OCP(\(\mu\)), the parameter \(\mu\) could represent some physical, natural or geometrical features. The present article investigates OCP(\(\mu\))s to quadratic optimization models constrained to linear time dependent PDE(\(\mu\)), in the form they are formulated in the papers of [F. Negri et al., Comput. Math. Appl. 69, No. 4, 319–336 (2015; Zbl 1421.49026); SIAM J. Sci. Comput. 35, No. 5, A2316–A2340 (2013; Zbl 1280.49046)]. The second section starts with the presentation of parabolic time dependent OCP(\(\mu\))s in saddle point formulation. Under some special assumptions and using an earlier “Brezzi theorem” the well-posedness of the saddle point structure in space-time formulation is proved. A discretized version of time dependent OCP(\(\mu\))s by means of an all-at-once space-time discretization procedure is obtained. The third section is devoted to the application of R0M (reduce order methods) approximations to time dependent OCP(\(\mu\)). The P0D-Galerkin algorithm is presented and it is extended to parametrized time dependent OCP(\(\mu\))s. In order to validate the performances of theoretical results, in the forth and fifth sections, two examples are presented and largely discussed: a time dependent boundary optimal control problem for Graetz flow and a time dependent distributed optimal control problem for Stokes equations, both examples having geometrical and phisical parametrizations. Numerical results for a time dependent OCP(\(\mu\)) for a Cavity Viscous Flow are shown in the sixth section. Conclusions, perspectives, next improvements and References with an impressive number of 60 titles, end the article.

MSC:
49N10 Linear-quadratic optimal control problems
49M25 Discrete approximations in optimal control
35J20 Variational methods for second-order elliptic equations
65N06 Finite difference methods for boundary value problems involving PDEs
90C20 Quadratic programming
Software:
RBniCS; FEniCS
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