Shape and Topological Derivatives via One Sided Differentiation of the Minimax of Lagrangian Functionals

Abstract

A standard approach to the minimization of an objective function in the presence of equality constraints in Mathematical Programming or of a state equation in Control Theory is the introduction of Lagrange multipliers or an adjoint state, that is, a linear penalization of the equality constraints or the state equation. The initial minimization problem is equivalent to the minimax of the associated Lagrangian. This approach can also be used to compute the one-sided directional derivative with respect to the control or the shape or topology of a family of sets. It is sufficient to consider a Lagrangian parametrized by a positive parameter t > 0 as t goes to 0. This involves the introduction of some forms of equation for the multiplier or the adjoint state at t > 0. The simplest one is the standard adjoint, but other forms of closely related adjoint such as the quasi adjoint in Serovaiskii (Mat Zametki 54(2):85–95, 159, 1993; translation in Math Notes 54(1–2):825–832, 1994) and Kogut et al. (Z Anal Anwend (J Anal Appl) 34(2):199–219, 2015) or the new averaged adjoint in Sturm (On shape optimization with non-linear partial differential equations, Doctoral thesis, Technische Universiltät of Berlin, 2014; SIAM J Control Optim 53(4):2017–2039, 2015) can be considered. In general, the various forms of adjoints coincide at t = 0. For instance, by using the averaged adjoint, the minimax problem need not be related to a saddle point as in Correa-Seeger (Nonlinear Anal Theory Methods Appl 9:13–22, 1985) and the so-called dual problem need not make sense.

In this paper, we sharpen the results of Sturm (On shape optimization with non-linear partial differential equations, Doctoral thesis, Technische Universiltät of Berlin, 2014; SIAM J Control Optim 53(4):2017–2039, 2015) which have been recently extended in Delfour and Sturm (J Convex Anal 24(4):1117–1142, 2017; Minimax differentiability via the averaged adjoint for control/shape sensitivity. In: Proceedings of the 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations, IFAC-PaperOnLine 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. In such a case, a non-differentiability can occur and only a one-sided directional derivative is expected even if the functions at hand are infinitely differentiable as was illustrated in the seminal paper of Danskin (SIAM J Appl Math 14(4):641–644) in 1966. Some examples for control and for shape and topological derivatives will be given.