# Control, Shape, and Topological Derivatives via Minimax Differentiability of Lagrangians

• Michel C. Delfour
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 29)

## Abstract

In Control Theory, the semidifferential of a state constrained objective function can be obtained by introducing a Lagrangian and an adjoint state. Then the initial problem is equivalent to the one-sided derivative of the minimax of the Lagrangian with respect to a positive parameter t as it goes to 0. In this paper, we revisit the results of Sturm (On shape optimization with non-linear partial differential equations. Doctoral thesis, Technische Universität of Berlin, 2014; SIAM J Control Optim 53(4):2017–2039, 2015) recently extended by Delfour and Sturm (J Convex Anal 24(4):1117–1142, 2017; Delfour and Sturm, Minimax differentiability via the averaged adjoint for control/shape sensitivity. In: Proc. of the 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations, IFAC-PaperOnLine, vol 49-8, pp 142–149, 2016) from the single valued case to the case where the solutions of the state/averaged adjoint state equations are not unique. New simpler conditions are given in term of the standard adjoint and extended to the multivalued case. They are applied to the computation of semidifferentials with respect to the control and the shape and the topology of the domain. The shape derivative is a differential while the topological derivative usually obtained by expansion methods is not. It is a semidifferential, that is, a one-sided directional derivative in the directions contained in the adjacent tangent cone obtained from dilatations of points, curves, surfaces and, potentially, microstructures (Delfour, Differentials and semidifferentials for metric spaces of shapes and geometries. In: Bociu L, Desideri JA, Habbal A (eds) System Modeling and Optimization. Proc. 27th IFIP TC7 Conference, CSMO 2015, Sophia-Antipolis, France. AICT Series, pp 230–239. Springer, Berlin, 2017; Delfour, J Convex Anal 25(3):957–982, 2018) by using the notion of d-dimensional Minkowski content. Examples of such sets are the rectifiable sets (Federer, Geometric measure theory. Springer, Berlin, 1969) and the sets of positive reach (Federer, Trans Am Math Soc 93:418–419, 1959).

## Keywords

Minimax One-sided differentiability Minimax Lagrangian Standard adjoint Control Shape Topological derivatives Rectifiability

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