Matrix Pontryagin principle approach to controllability metrics maximization under sparsity constraints

We define a mapping $\psi_{nm}:\mathbb{R}^{n\times m}\to\mathbb{R}^{nm}$ by

where $X_{i}\in\mathbb{R}^{n}$ denotes the $i$th column of a matrix $X\in\mathbb{R}^{n\times m}$. From Athans (1967), $\psi_{nm}$ is a regular linear mapping (hence $\psi_{nm}^{-1}$ exists), and preserves the inner product. Then problem (9) is equivalent to

where $x=\psi_{nm}(X)$, $u=\psi_{pq}(U)$, and $l_{f},f,x_{0},e,\omega$ corresponds to $L_{f},F,X_{0},E,\Omega$ respectively. We define the Hamiltonian function $h:\mathbb{R}^{nm}\times\mathbb{R}^{nm}\times\mathbb{R}^{pq}\to\mathbb{R}$ associated to problem (16) by $h(x(t),p(t),u(t))=p(t)^{\top}f(x(t),u(t))$ and denote the local minimizer for problem (16) by ($x^{*}(t)$, $u^{*}(t)$). Then there exists an arc $p:[0,T]\rightarrow\mathbb{R}^{nm}$ and a scalar $\eta$ equal to 0 or 1 satisfying the following conditions (Pontryagin’s Maximum Principle (Clarke (2013))):

• •

  the nontriviality condition:

  $$(\eta,p(t))\neq 0\quad^{\forall}t\in[0,T],$$

  (17)

• •

  the transversality condition:

  $$-p(T)\in\eta\nabla l_{f}(x^{*}(T))+N_{e}^{L}(x^{*}(T)),$$

  (18)

• •

  the adjoint equation for almost every $t\in[0,T]$

  $$-\dot{p}(t)=D_{x}h(x^{*}(t),p(t),u^{*}(t)),$$

  (19)

• •

  the maximum condition for almost every $t\in[0,T]$

  $$h(x^{*}(t),p(t),u^{*}(t))=\sup_{u\in\omega}h(x^{*}(t),p(t),u).$$

  (20)

Since $\psi_{nm}^{-1}$ exists, we obtain the Hamiltonian function associated to $h(x^{*}(t),p(t),u^{*}(t))$ as follows:

which satisfies (11), (12), (13), and (14), where $X^{*}=\psi_{nm}^{-1}(x^{*})$, $U^{*}=\psi_{pq}^{-1}(u^{*})$, $P=\psi_{nm}^{-1}(p)$. This completes the proof. ∎

Fix any $\hat{A}\in E$ and $\hat{a}=\psi_{nn}(\hat{A})$, where $\psi_{nn}$ is from (15). Then we obtain a set $e$ satisfying $\hat{a}\in e$ as follows:

where $\alpha^{\prime}=\psi_{nn}(\alpha)$ and $\mathcal{I}^{\prime}\subset\mathbb{N}^{n^{2}}$ is a set corresponding to $\mathcal{I}$. Take any $\gamma^{\prime}\in e$, then we have

for all $\delta^{\prime}\in N_{E}^{L}(\gamma^{\prime})$ (Ikeda and Kashima (2022)). Finally, we obtain (23), (24), (25) where $\delta=\psi_{nn}^{-1}(\delta^{\prime})$ and $\gamma=\psi_{nn}^{-1}(\gamma^{\prime})$. ∎

We first reformulate Problem 2 into a form to which Lemma 1 is applicable. The value $\|v_{j}\|_{L^{1}}$ is equal to the final state $y_{j}(T)$ of the system $\dot{y_{j}}(t)=v_{j}(t)$ with $y_{j}(0)=0$. Define $Y(t)\triangleq{\rm diag}{\left(y(t)\right)}$ and matrices $X(t)$, $\bar{V}(t)$, $\bar{A}$, $\bar{B}$ by

Then, Problem 2 is equivalently expressed as follows:

where $L_{f}(X(T))=K(G_{c}(T))$, $E=\{X(T):y_{j}(T)\leq\alpha_{j}\quad^{\forall}j\in\{1,2,\dots,p\}\}$, $\Omega=\{v(t):v(t)\in[0,1]^{p},\|v(t)\|_{l^{1}}\leq\beta\}$. This is an optimal control problem to which Lemma 1 is applicable. We define the Hamiltonian function $H$ associated to problem (30) by

We define two matrices as follows:

Then ($X^{*}(t)$, $\bar{V}^{*}(t)$) is the local minimizer of problem (30) because of the equivalence between Problem 2 and problem (30), and there exists a scalar $\eta$ equal to 0 or 1 and a matrix $P:[0,T]\rightarrow\mathbb{R}^{n\times n}$ satisfying the conditions (11), (12), (13), (14). It follows from (13) that

which leads to

where

with $P^{(11)}(t)\in\mathbb{R}^{n\times n}$ and $P^{(22)}(t)\in\mathbb{R}^{p\times p}$. Note that

by (12), then we have

from Lemma 2. Substituting these into (32), we get

Then, we have

It follows from (14) that

We here claim that $\eta=1$. Indeed, if $\eta=0$, $P^{(22)}(T)\neq O_{p\times p}$ follows from (11), i.e., there exists some $j$ that satisfies

Hence, from (40) and (41), we have $v_{j}^{*}(t)=0$ for all $t\in[0,T]$, i.e., $y_{j}^{*}(T)=\|v_{j}^{*}\|_{L^{1}}=0$. This contradicts to (41). Thus, $\eta=1$. From the assumption, it is easy to verify that

1. 1)

   we have $q_{j}(t)+P_{j,j}^{(22)}(T)\neq 0$ almost everywhere for all $j=\{1,2,\dots,p\}$,

2. 2)

   there exists $j_{k}$ $:$ $[0,T]\rightarrow\{1,2,\dots,p\},k=1,2,\dots,p$, such that

   $$q_{j_{1}(t)}(t)+P_{j_{1}(t),j_{1}(t)}^{(22)}(T)>\cdots>q_{j_{p}(t)}(t)+P_{j_{p}(t),j_{p}(t)}^{(22)}(T)$$

   almost everywhere.

Hence, we find

for almost every $t\in[0,T]$, where

This completes the proof. ∎

Denote any solution of Problem 2 by $\hat{v}\in\mathcal{V}_{1}^{*}$. It follows from Theorem 1 that $\hat{v}(t)\in\{0,1\}^{p}$ almost everywhere. Note that the null set $\cup_{j=1}^{p}\{t\in[0,T]:\hat{v}_{j}(t)\notin\{0,1\}\}$ does not affect the cost, and hence we can adjust the variables so that $\hat{v}(t)\in\{0,1\}^{p}$ on $[0,T]$, without loss of the optimality. We have

for all $j$. Since $\hat{v}\in\mathcal{V}_{1}$, we have $\|\hat{v}(t)\|_{l^{0}}\leq\beta$ and $\|\hat{v}_{j}\|_{L^{0}}\leq\alpha_{j}$ for all $t$ and $j$. Thus, $\hat{v}\in\mathcal{V}_{0}$. Then,

where the first relation follows from $\hat{v}\in\mathcal{V}_{0}$, the second relation follows from $\mathcal{V}_{0}\subset\mathcal{V}_{1}$, and the last relation follows from $\hat{v}\in\mathcal{V}_{1}^{*}$. Hence, we have

which implies $\hat{v}\in\mathcal{V}_{0}^{*}$. Hence, $\mathcal{V}_{1}^{*}\subset\mathcal{V}_{0}^{*}$ and $\mathcal{V}_{0}^{*}$ is not empty.

Next, take any $\tilde{v}\in\mathcal{V}_{0}^{*}$. Note that $\tilde{v}\in\mathcal{V}_{1}$, since $\mathcal{V}_{0}^{*}\subset\mathcal{V}_{0}\subset\mathcal{V}_{1}$. In addition, it follows from (44) that $J(\tilde{v})=J(\hat{v})$. Therefore, $\tilde{v}\in\mathcal{V}_{1}^{*}$, which implies $\mathcal{V}_{0}^{*}\subset\mathcal{V}_{1}^{*}$. This gives $\mathcal{V}_{0}^{*}=\mathcal{V}_{1}^{*}$. ∎