The interval turnpike property for adjoints

Abstract

In this work we derive an interval turnpike result for adjoints of finite-and infinite-dimensional nonlinear optimal control problems under the assumption of an interval turnpike on states and controls. We consider stabilizable dynamics governed by a generator of a semigroup with finite-dimensional unstable part satisfying a spectral decomposition condition and show the desired turnpike property under continuity assumptions on the first-order optimality conditions. We further give stronger estimates for analytic semigroups and provide a numerical example with a boundary controlled semilinear heat equation to illustrate the results.

Linear turnpike theorem

“…(iii) The family of optimal control problems (1)-( 2), indexed by T > 0, is strictly dissipative at the optimal static point (ȳ, ū) with respect to the supply rate function w defined by (9), with a storage function S (satisfying (10)) that is bounded on M 0 and M 1 .…”

“…The turnpike result of [29] has been extended to the infinite-dimensional setting in [27,28,10,17]. One can find many turnpike results in the literature (see [5,11,16,15,20,21,32,33], and see historical references in [29]), but the specificity of the above result is that the turnpike property is established as well for the adjoint vector: as explained in [29], this is particularly important in view of deriving appropriate initializations in numerical computation methods (direct methods or shooting methods, see [26]).…”

“…This property follows from the strict dissipativity assumption (iii) and from the controllability assumption (iv). Indeed, the strict dissipativity inequality (10) implies that…”