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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
RBniCS; FEniCS
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