--- title: Sparse Optimal Control url: https://www.emergentmind.com/topics/sparse-optimal-control type: topic --- # Sparse Optimal Control Sparse optimal control denotes a family of optimal control formulations in which some notion of sparsity is optimized together with stability or performance. In the literature, the sparse object may be the time support of the control, the set of active actuator or sensor channels, the spatial or spatio-temporal support of a distributed control, or the frequency support of a time–frequency representation. Correspondingly, sparse optimal control appears as minimum-support or “maximum hands-off” control, as $\ell^1$- or group-regularized controller synthesis, as PDE-constrained control with sparsity-inducing penalties, and as finite-actuator control of continuum or mean-field models [1412.5707] [2409.09596] [1507.00768]. ## 1. Foundational formulations and equivalence principles A classical finite-dimensional formulation considers the linear time-invariant single-input system \[ \frac{d\mathbf{x}(t)}{dt} = A \mathbf{x}(t) + B u(t), \] with fixed horizon $T>0$, terminal condition $\mathbf{x}(T)=\mathbf{0}$, and amplitude constraint $\|u\|_\infty \le 1$. The sparse objective is the Lebesgue measure of the control support, \[ \|u\|_0 \triangleq m(\mathrm{supp}(u)), \] leading to \[ P_0:\quad \min_{u \in U(\boldsymbol{\xi})} \ \|u\|_0. \] The associated minimum-fuel problem is \[ P_1:\quad \min_{u \in U(\boldsymbol{\xi})} \ \|u\|_1. \] Under the normality assumption, the two problems coincide: if $P_1$ is normal and admits an optimal control, then \[ U_0^*(\boldsymbol{\xi}) = U_1^*(\boldsymbol{\xi}), \qquad \|u_0\|_0 = \|u_1\|_1, \] and therefore $V_0(\boldsymbol{\xi}) = V_1(\boldsymbol{\xi})$ on the reachable set $R(T)$. In that regime, sparse optimal controls can be computed through the convex $L^1$ problem rather than directly through the nonconvex $L^0$ problem. The corresponding Pontryagin structure is bang-off-bang: \[ u^*(t) = -\mathrm{dez}\bigl(B^{\top}\mathbf{p}^*(t)\bigr), \] where the switching function is $B^\top \mathbf{p}^*(t)$ and $\mathrm{dez}$ is the dead-zone map. Under controllability and nonsingular $A$, the value function $V_0$ is continuous on $R(T)$ [1412.5707]. Infinite-horizon sparse optimal control extends this viewpoint to discounted problems with running cost \[ J(x,u)=\int_0^\infty e^{-\lambda s}\Big(\ell_1(y(s))+\gamma\|u(s)\|_p^p\Big)\,ds, \qquad 0\lambda\gamma$ [1602.08931]. ## 2. Convex co-design of sparse controllers, actuators, and sensors A distinct line of work treats sparse optimal control as a controller-synthesis problem for continuous-time LTI plants with prescribed $\mathcal H_2$ or $\mathcal H_\infty$ performance. The plant is written as \[ \dot{x} = A x + B u + B_1 w,\qquad y = C_y x + D_{yw} w,\qquad z = C_z x + D_z u + D_{zw} w, \] and two controller classes are considered: static full-state feedback $u=Kx$ and dynamic output feedback \[ \dot{x}_K = A_K x_K + B_K y,\qquad u = C_K x_K + D_K y. \] Sparsity is induced in two ways. Channel-wise actuator sparsity is promoted by minimizing a weighted $\ell_1$ norm of bounds on the disturbance-to-control $\mathcal H_2$ norms of individual channels, while structural sparsity is induced by $\ell_1/\ell_2$ penalties on rows of $[\hat C_K\ \hat D_K]$ and columns of $[\hat B_K\ \hat D_K]$. In the static full-state case, the change of variables $X \succ 0$, $W:=KX$ converts bilinear synthesis constraints into semidefinite programs. In the output-feedback case, Scherer’s transformed variables $\hat A_K,\hat B_K,\hat C_K,\hat D_K$ and Lyapunov partitions are used, together with realizability constraints such as $D_K D_{yw}=0$ and, for $\mathcal H_2$, $D_z=0$ when $D_{yw}\neq 0$ [2409.09596]. The crucial structural result is sparsity preservation: if $[\hat C_K\ \hat D_K]$ is row-sparse and $[\hat B_K;\hat D_K]$ is column-sparse, then the reconstructed controller matrices $[C_K\ D_K]$ and $[B_K;D_K]$ inherit the same actuator and sensor sparsity. The resulting problems are convex SDPs, so the synthesized solutions are globally optimal within the relaxed formulation class. The paper’s tensegrity-wing study makes the architecture implications explicit. For $\gamma_0=0.1$, sparse full-state and sparse output-feedback designs select primarily cables $3$, $4$, and $7$, with slight use of cable $6$ in output feedback. Under structural sparsity, the $\mathcal H_2$ design selects actuators $3,4,6,7$ and sensors $4,6,10,12$, whereas the $\mathcal H_\infty$ design selects actuators $3,4,6,7$ and sensors $6,10,11,12$. Tightening channel bounds $\gamma_i \le \gamma_{i,\max}$ forces additional actuators into the design, exposing the performance–magnitude–sparsity trade-off [2409.09596]. ## 3. PDE-constrained sparse control In PDE-constrained settings, sparsity is usually enforced by an $L^1$ term in the control cost. For fractional diffusion with the spectral fractional operator $\mathcal L^s$, the cost \[ J(u,z) = \frac{1}{2}\,\|u - u_d\|_{L^2(\Omega)}^2 + \frac{\sigma}{2}\,\|z\|_{L^2(\Omega)}^2 + \nu\,\|z\|_{L^1(\Omega)} \] leads to the pointwise optimality condition \[ \bar z(x') = \operatorname{Proj}_{[a,b]}\!\left( -\frac{1}{\sigma}\,S_{\nu}\big(\bar p(x')\big) \right), \qquad \bar z(x')=0 \iff |\bar p(x')|\le \nu. \] Using the Caffarelli–Silvestre extension on the truncated cylinder and piecewise-constant controls with tensor-product finite elements for the state, the a priori control and state errors scale like \[ |\log(\#\mathscr{T}_Y)|^{\,2s}\,(\#\mathscr{T}_Y)^{-1/(n+1)} \] up to the norms specified in the analysis [1704.01058]. For semilinear elliptic equations with the integral fractional Laplacian, the same thresholding structure persists. With \[ J(y,u) = \int_\Omega L(x,y)\,dx + \frac{\alpha}{2}\|u\|_{L^2(\Omega)}^2 + \beta\|u\|_{L^1(\Omega)}, \] the first-order condition becomes \[ u^*(x) = \Pi_{[a,b]}\!\left(-\frac{1}{\alpha}\big(p^*(x)+\beta\,\eta^*(x)\big)\right), \qquad \eta^*(x) = \Pi_{[-1,1]}\!\left(-\frac{1}{\beta}p^*(x)\right), \] hence \[ u^*(x)=0 \iff |p^*(x)|\le \beta. \] The analysis includes second-order conditions on a critical cone and finite-element error estimates for both fully discrete and semidiscrete schemes [2312.08335]. Other distributed PDE models exhibit more structured sparsity. For the viscous Camassa–Holm equations, three convex sparsity functionals are treated: \[ j_1(u)=\|u\|_{L^1(Q)^3},\qquad j_2(u)=\|u\|_{L^2(0,T;L^1(\Omega)^3)},\qquad j_3(u)=\|u\|_{L^1(\Omega;L^2(0,T)^3)}. \] They induce, respectively, pointwise spatio-temporal sparsity, time-sparse thresholds, and spatial group sparsity. The corresponding control laws imply \[ |\bar\lambda_i(x,t)|\le \kappa \Rightarrow \bar u_i(x,t)=0 \] for $j_1$, \[ |\bar\lambda_i(x,t)|\le \kappa\,\bar\sigma_i(t) \Rightarrow \bar u_i(x,t)=0 \] for $j_2$, and \[ \|\bar\lambda_i(x,\cdot)\|_{L^2(0,T)}\le \kappa \iff \|\bar u_i(x)\|_{L^2(0,T)}=0 \] for $j_3$. As $\kappa\to 0$, the optimal controls converge to those of the nonsparse problem, with rates $O(\kappa)$ for $j_1$ and $j_3$, and $O(\kappa^{2/3})$ for $j_2$ [2604.02724]. For viscous Cahn–Hilliard systems with logarithmic potential, sparsity enters through \[ \mathcal{J}(\varphi,u)=\frac{b_1}{2}\int_Q|\varphi-\varphi_Q|^2\,dx\,dt +\frac{b_2}{2}\int_\Omega|\varphi(T)-\varphi_\Omega|^2\,dx +\frac{b_3}{2}\int_Q|u|^2\,dx\,dt +K\,\|u\|_{L^1(Q)}. \] The first-order condition is \[ \int_Q\big(r^*+b_3 u^*+K\,\chi^*\big)\,(u-u^*)\,dx\,dt \ge 0, \] with $\chi^*\in \partial \|u^*\|_{L^1(Q)}$. If $0\in[\underline u,\overline u]$, then \[ |r^*(x,t)|