Relaxation Methods for Mixed-Integer Optimal Control of Partial Differential Equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments

k-Extreme Points in Symmetric Spaces of Measurable Operators

Let $\mathcal{M}$ be a semifinite von Neumann algebra with a faithful, normal, semifinite trace $\tau$ and $E$ be a strongly symmetric Banach function space on $[0,\tau(1))$. We show that an operator $x$ in the unit sphere of $E\left(\mathcal{M},\tau\right)$ is $k$-extreme, $k\in\mathbb N$, whenever its singular value function $\mu(x)$ is $k$-extreme and one of the following conditions hold (i) $\mu(\infty,x)=\lim_{t\to\infty}\mu(t,x)=0$ or (ii) $n(x)\mathcal{M} n(x^*)=0$ and $|x|\geq \mu(\infty,x)s(x)$, where $n(x)$ and $s(x)$ are null and support projections of $x$, respectively. The converse is true whenever $\mathcal{M}$ is non-atomic. The global $k$-rotundity property follows, that is if $\mathcal{M}$ is non-atomic then $E$ is $k$-rotund if and only if $E\left(\mathcal{M},\tau\right)$ is $k$-rotund. As a consequence of the noncommutive results we obtain that $f$ is a $k$-extreme point of the unit ball of the strongly symmetric function space $E$ if and only if its decreasing rearrangement $\mu(f)$ is $k$-extreme and $|f|\geq \mu(\infty,f)$. We conclude with the corollary on orbits $\Omega(g)$ and $\Omega'(g)$. We get that $f$ is a $k$-extreme point of the orbit $\Omega(g)$, $g\in L_1+L_{\infty}$, or $\Omega'(g)$, $g\in L_1[0,\alpha)$, $\alpha<\infty$, if and only if $\mu(f)=\mu(g)$ and $|f|\geq \mu(\infty,f)$. From this we obtain a characterization of $k$-extreme points in Marcinkiewicz spaces.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00020-014-2206-