Optimization and discrete approximation of sweeping processes with controlled moving sets and perturbations

Controlled polyhedral sweeping processes: existence, stability, and optimality conditions

“…Note that results of this type in the W 1,2 controltrajectory framework we obtained in [31,32,33] for various types of sweeping processes under appropriate assumptions in separable Hilbert spaces. Similar preservation results of class W 1,2 were established in [9] in finite dimensions under the strong uniform Slater condition (2.12) reducing to (2.11) as we now know. Observe also that results of this type in class of W 1,1 were derived in [13,12] for polyhedral sweeping processes in finite-dimensional spaces under essentially stronger qualification conditions than (2.11) used in what follows.…”

“…However, this does not make much sense when the moving set is an unbounded polyhedron. The W 1,2 -preserving existence and uniqueness results for moving polyhedra were obtained by Tolstonogov [31,32,33] and more recently in [9] under certain qualification conditions in Hilbert and finite-dimensional settings; see more discussions in Section 3. Here we develop a novel approach involving the truncation of polyhedra and deriving refined error bounds.…”

“…As mentioned, the existence of the unique Lipschitz continuous trajectory x(•) of the Cauchy problem for the polyhedral sweeping process in (1.2) generated by the given canonical control pair (ū(•), b(•)) follows from Theorem 2. Now we are in a position to deduce the claimed assertions from [9,Theorem 4.1] under the BV assumption on u(•), ḃ(•), and ẋ(•). Indeed, the qualification condition (H4) from [9, Theorem 4.1] is equivalent to the uniform Slater condition (2.11) by our new result obtained in Proposition 1.…”

“…Sweeping differential inclusions are highly discontinuous, and the machinery of Lipschitzian variational analysis is not applicable in the sweeping framework. Further developments of this method in various sweeping control settings can be found in [2,9,8,10,13,14] and the references therein. However, neither these publications, nor those of [5,15,35] exploring other approaches to deriving optimality conditions in different models of sweeping optimal control address additional endpoint constraints x(T ) ∈ Ω on sweeping trajectories.…”

“…The third topic we investigate here concerns an optimal control problem for the sweeping process in (1.1) and (1.2) under the additional pointwise equality constraint on the u-component of controls and geometric endpoint constraint x (u,b) ∈ Ω on trajectories. Optimal control theory for sweeping processes, with addressing the main issue of deriving necessary optimality conditions, has been started rather recently in [11] and then has been extensively developed in subsequent publications (see, e.g., [2,5,8,9,10,12,13,14,15,35] and the references therein), which did not concern however systems with endpoint constraints. Problems of sweeping optimal control, that are governed by discontinuous dif-ferential inclusions with intrinsic pointwise and irregular state constraints, constitute one of the most challenging class in modern control theory.…”