Regularization of Linear-Quadratic Control Problems with L 1 -Control Cost

. We analyze L 2 -regularization of a class of linear-quadratic optimal control problems with an additional L 1 -control cost depending on a parameter β . To deal with this nonsmooth problem we use an augmentation approach known from linear programming in which the number of control variables is doubled. It is shown that if the optimal control for a given β ∗ ≥ 0 is bang-zero-bang, the solutions are contin-uous functions of the parameter β and the regularization parameter α . Moreover we derive error estimates for Euler discretization.


Introduction
The regularization of optimal control problems by a L 2 -term α 2 u 2 L 2 is often used in order to get a smoother optimal control. In this cases α can be viewed as a regularization parameter and one is interested in the question how the solutions depend on this parameter. For the special case that the control variable appears linearly in the control problem and the optimal control without regularization (α = 0) has bang-bang structure this question has been investigated in Deckelnick/Hinze [1] for a class of elliptic control problems and in Alt/Seydenschwanz [2] for a general class of linear-quadratic control problems governed by ordinary differential equations.
Maurer/Vossen [3] investigate first order necessary and second order sufficient optimality conditions for a class of nonlinear control problems involving a L 1 -term in the cost functional, where the parameter β is kept fixed. They also propose some numerical algorithms for the solution of such problems. Sakawa [4] also considers a special numerical algorithm for a fixed parameter β > 0. Stadler [5] and Casas et al. [6,7] investigate classes of elliptic control problems with a L 1 -term in the cost functional, which is interpreted as a regularization term. They derive results on the dependence of the solutions on the parameter β and error estimates for discretizations, but an additional L 2 -regularization term with fixed parameter α is used in order to get smoother solutions. In Wachsmuth/Wachsmuth [8] the dependence of solutions of a class of elliptic control problems on the regularization parameter α is studied while the parameter β is kept fix.
Results for the dependence of the solutions on the parameter β and error estimates for discretizations for a general class of linear-quadratic control problems governed by ordinary differential equations have been recently derived in [9]. In the present paper, we investigate the regularization of such control problems and the dependence of solutions on the parameter β and the regularization parameter α assuming that for a fixed parameter β * the corresponding optimal control is of bang-zero-bang type.

Problem formulation
, we consider the following family of L 2 -regularized linear-quadratic control problems with L 1 -control cost depending on the parameters α ≥ 0 and β ≥ 0: where f α,β is a linear-quadratic cost functional with an additional nonsmooth L 1 -term defined by Here, u(t) ∈ R m is the control, and x(t) ∈ R n is the state of the system at time t, where t ∈ [0, t f ]. Further Q ∈ R n×n is a symmetric and positive semidefinite matrix, q ∈ R n , and the functions W : The matrices W (t) are assumed to be symmetric and positive semidefinite, and the set U ∈ R m is defined by lower and upper bounds, i.e.
where all inequalities are to be understood componentwise. While the regularization term α 2 u 2 L 2 leads to a smooth optimal control for α > 0 the term β u L 1 may be interpreted as both a regularization or some (nonsmooth) L 1 -control cost. We are interested in the behavior of a solution u α,β of Problem (PQ α,β ) depending on both parameters α and β.

Optimality conditions
We denote by the set of admissible controls, and by the feasible set of (PQ α,β ). Since U is nonempty, the feasible set F is nonempty, too. And since U is bounded, it follows thatẋ is bounded for any feasible pair (x, u) ∈ F, and therefore F ⊂ X. Moreover, there is some constant c such that Since the feasible set F is nonempty, closed, convex and bounded, and the cost functional is convex and continuous, a minimizer ( and the minimum principle hold a.e. on [0, t f ] (compare e.g. [11,Theorem 10.47] or [3, Sect. 2]).
Provided α = 0 we are able to evaluate the minimum principle (2) in more detail (compare [3] and [9]) and obtain where ξ β (t) := B(t) T λ 0,β (t) + r(t). If we assume that the set of switching times is finite, then by (3) the i-th component of the optimal control has a bang-zerobang structure.

Problem transformation
In common with [3] and [9] we formulate a transformed problem (TQ α,β ) in order to study the dependence of the optimal control on the parameters α and β. This is a well known augmentation approach from linear programming wherewith we obtain a linear-quadratic control problem with smooth cost functional (see e.g. [12]).
Introducing new controls v ∈X 2 := L ∞ (0, t f ; R 2m ) and using the matrix where B(t) := B(t)M . There are new box constraints for the controls, and f α,β is a linear-quadratic cost functional: With the same argumentation as above for Problem (PQ α,β ) we are able to show that a minimizer of Problem (TQ α,β ) exists. We denote the set of admissible controls by and the feasible set of Problem (TQ α,β ) by T ⊂ X 1 ×X 2 , where Although Problem (TQ α,β ) admits controls with components v 2i−1 , v 2i being positive simultaneously, such controls cannot be optimal (see [3,Sect. 4 The optimality conditions also prove this result. By (5) and v(t) ≥ 0 we now are able to simplify which nicely shows, that a L 1 -or L 2 -regularization of the original problem implies the same regularization of the transformed problem. We finally introduce the minimum principle of Problem (TQ α,β ) where with e := (1, . . . , 1) T ∈ R 2m . The adjoint equation (1) as well as the adjoint variables λ α,β do not change im comparison to Problem (PQ α,β ). A detailed discussion of the optimality conditions can be found in [3] and [9].
Let I(s j ) := {1 ≤ i ≤ 2m | σ 0,β * i (s j ) = 0} be the set of active indices for the components of the switching function. In order to get stability of the bang-bang structure under perturbations we need an additional assumption (compare [14]): (B2) The functions B and r are differentiable,Ḃ andṙ are Lipschitz continuous, and there existsσ > 0 such that Remark 2. Assumption (B2) can be slightly relaxed (see e.g. [15,9]).
Remark 3. By Theorem 1 we also obtain estimates for the optimal states and for the optimal controls u α,β of the original problem (PQ α,β ), by using the matrix (4) and the relation (5) between u α,β and v α,β If we choose some β in a sufficiently small neighborhood of β * this result can even be improved.
Proof. We use [9, Theorem 6.3, Remark 10], which proved the local Lipschitzcontinuity of the optimal control depending on β, where the constantc is independent of β: In addition to this we are able to extend the result of [2, Theorem 4.1] using the problem transformation introduced in Sect. 4 and obtain with some constantc independent of α. Together (15) and (16) lead to which implies (14).

Discretization
For the numerical solution of Problem (PQ α,β ) we use the Euler discretization scheme described in [15] and [9]. Given a natural number N and let h N = t f /N be the meshsize, we approximate the cost functional f α,β by and Problem (PQ α,β ) by Remark 4. Note that analogously to [9] we solve a transformed discretized problem (compare also Sect. 4) to compute the solution of Problem (PQ α,β ) numerically.
with some constantc β independent of β, which implies the assertion.