Boundary control design for conservation laws in the presence of measurement disturbances

Abstract

Boundary feedback control design for systems of linear hyperbolic conservation laws in the presence of boundary measurements affected by disturbances is studied. The design of the controller is performed to achieve input-to-state stability (ISS) with respect to measurement disturbances with a minimal gain. The closed-loop system is analyzed as an abstract dynamical system with inputs. Sufficient conditions in the form of dissipation functional inequalities are given to establish an ISS bound for the closed-loop system. The control design problem is turned into an optimization problem over matrix inequality constraints. Semidefinite programming techniques are adopted to devise systematic control design algorithms reducing the effect of measurement disturbances. The effectiveness of the approach is extensively shown in several numerical examples.

Appendix A Ancillary results and definitions

Definition A1

[15] Let $\mathcal {Z}$ be a Hilbert space. The function $\mathscr {T}:\mathbb {R}_{\ge 0}\rightarrow \mathscr {L}(\mathcal {Z}, \mathcal {Z})$ is a $\mathcal {C}_0$-semigroup on $\mathcal {Z}$ if it satisfies the following properties:

• (a) For all $t, s\in \mathbb {R}_{\ge 0}$, $\mathscr {T}(t+s)= \mathscr {T}(t) \mathscr {T}(s)$

• (b) $\mathscr {T}(0)=\mathbf {I}$

• (c) For all $z_0\in \mathcal {Z}$, $\displaystyle \lim _{t\rightarrow 0^+}\Vert \mathscr {T}(t)z_0-z_0\Vert =0$

Definition A2

[15] Let $\mathcal {Z}$ be a Hilbert space and $\mathcal {A}:{\text {dom}}\mathcal {A}\rightarrow \mathcal {Z}$. We say that $\mathcal {A}$ generates a $\mathcal {C}_0$-semigroup $\mathscr {T}$ on $\mathcal {Z}$ if for all $z\in {\text {dom}}\mathcal {A}$

$$\begin{aligned} {\mathcal {A}}z=\lim _{t\rightarrow 0^+}\frac{1}{t}\left( \mathscr {T}(t)-\mathbf {I}\right) z \end{aligned}$$

$\square$

Definition A3

([13]) Let X and Y be linear normed spaces, U be an open subset of X, $f:U\rightarrow Y$, and $x\in U$. We say that f is Fréchet differentiable at x if there exists $L\in \mathscr {L}(X,Y)$ such that

$$\begin{aligned} \lim _{h\rightarrow 0}\frac{\Vert f(x+h)-f(x)-Lh\Vert _Y}{\Vert h\Vert _X}=0 \end{aligned}$$

In particular L is the Fréchet derivative of f at x and is denoted by Df(x). When $X=\mathbb {R}$, we denote

$$\begin{aligned} \dot{f}(x)=\lim _{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} \end{aligned}$$

$\square$

Lemma A1

Let $\psi \in \mathcal {C}^0([0, 1]; \mathbb {R}), G\in \mathbb {S}_+^n$ and $\mathcal {L}^2((0,1); \mathbb {R}^{n_p})$ be endowed with its standard inner product. Consider the following functional

$$\begin{aligned} \begin{aligned} V:&\mathcal {L}^2((0,1); \mathbb {R}^{n_p})\rightarrow \mathbb {R}\\&X\mapsto V(X):=\int _0^1\psi (z)\langle G X(z), X(z)\rangle _{\mathbb {R}^{n_p}} \mathrm{d}z \end{aligned} \end{aligned}$$

Then, V is Fréchet differentiable on $\mathcal {L}^2((0,1); \mathbb {R}^{n_p})$ and in particular for each $X, h\in \mathcal {L}^2((0,1); \mathbb {R}^{n_p})$

$$\begin{aligned} \mathrm{DV}(X) h=2\langle \psi G X, h\rangle _{\mathcal {L}^2((0,1); \mathbb {R}^{n_p})} \end{aligned}$$

Proof

For any $X, h\in \mathcal {L}^2((0,1); \mathbb {R}^{n_p})$, one has

$$\begin{aligned} \begin{aligned} V(X+h)-V(X)=&\int _0^1\psi (z)\left( \langle h(z), Gh(z) \rangle _{\mathbb {R}^{n_p}}\right. \\&+\left. 2 \langle X(z), Gh(z) \rangle _{\mathbb {R}^{n_p}} \right) \mathrm{d}z\\&\le \lambda _{\max }(G)\Vert \psi \Vert _{\mathcal {L}^\infty ((0,1);\mathbb {R})}\Vert h\Vert ^2_{\mathcal {L}^2((0,1); \mathbb {R}^{n_p})}\\&+2 \langle \psi GX, h\rangle _{\mathcal {L}^2((0,1); \mathbb {R}^{n_p})} \end{aligned} \end{aligned}$$

Thus, it follows that

$$\begin{aligned} \begin{aligned} \lim _{\displaystyle \Vert h\Vert _{\mathcal {L}^2((0,1); \mathbb {R}^{n_p})}\rightarrow 0}\frac{\vert V(X+h)-V(X)-2 \langle X, \psi Gh\rangle _{\mathcal {L}^2((0,1);\mathbb {R}^{n_p})}\vert }{\Vert h\Vert _{\mathcal {L}^2((0,1); \mathbb {R}^{n_p})}}=0 \end{aligned} \end{aligned}$$

This concludes the proof. $\square$