# Linear-Quadratic Control Problems with $\boldsymbol{L^1}$-Control Cost

Christopher Schneider*

We analyze a class of linear-quadratic optimal control problems with additional $L^1$-control cost. These are optimization problems with nonsmooth cost functional: $\begin{array}{lll} &\min\; &\displaystyle\frac{1}{2} x(t_f)^\top Q x(t_f)+\displaystyle\int_0^{t_f} \! \frac{1}{2} x(t)^\top W(t)x(t) +w(t)^\top x(t)+r(t)^\top u(t) \,\mathrm dt + \beta\, \|u\|_{L^1}\\ &\rm{s.\,t.}\\ &&\dot x(t) = A(t)x(t) + B(t)u(t) \quad \text{a.e. on } [0,t_f]\,,\\ &&x(0) = a\,,\\ &&u(t) \in \left\{u\in\mathbb R^m \mid b_\ell \leq u \leq b_u\right\} \quad \text{a.e. on } [0,t_f]\,. \end{array}$ To deal with the nonsmooth problem we use an augmentation approach (see {[3]}) in which the number of control variables is doubled. It is shown that if the optimal control for a given $\bar\beta\geq 0$ is bang-bang, the solutions are continuous functions of the parameter $\beta$. We also show that the optimal controls for $\bar\beta$ and a $\beta$ with $|\beta-\bar\beta|$ sufficiently small coincide except on a set of measure $\mathcal O(\beta)$. \newline Since the minimum principles give the same results for both the original problem and the augmented one we use the Euler discretization to solve the augmented problem. Then we can refer to {[1]} for error bounds of the approximation. We also test other methods to solve the nonsmooth problem and then compare the results. \begin{thebibliography}{99} \bibitem{baier} W.~Alt, R.~Baier, M.~Gerdts and F.~Lempio, \emph{Error bounds for Euler approximation of linear-quadratic control problems with Bang-Bang solutions}, Numerical Algebra, Control and Optimization \textbf{2} (2012), 547--570. \bibitem{altschneider} W.~Alt, C.~Schneider, \emph{Linear-Quadratic Control Problems with $L^1$-Control Cost}, submitted. \bibitem{vossenmaurer} G.~Vossen and H.~Maurer, \emph{On $L^1$-Minimization in optimal control and applications to robotics}, Optimal Control Applications and Methods \textbf{27} (2006), 301--321. \end{thebibliography}

Mathematics Subject Classification: 49K15 49M25 49N10

Keywords: Optimal Control; Bang-Bang Control; $L^1$-Minimization; Nonsmooth Analysis; Regularization, Discretization

Minisymposion: Nonsmooth Optimization