User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Stability of Linear Delayed Differential Equations

The System


\begin{align} \dot x(t)&=Ax(t)+\sum_{i=1}^{L}A_ix(t-\tau_i),\\ \end{align}$$ where $$ x(t) \in \mathbb{R}^n $$ and $$ \tau_i >0$$.

The Data
The matrices $$ A,\{A_i.\tau_{i}\}_{i=1}^L $$.

The LMI:
Solve the following LMIP

\begin{align} &\text{Find} \{P \succ 0, P_1 \succ 0, \dots,P_L \succ 0\} :\\ &\quad  s.t. \begin{bmatrix} A^\top P +PA+\sum_{i=1}^L P_i & PA_1 & \dots & PA_L \\ A^\top_1P &  -P_1 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ A^\top_L P & 0 & \dots & -P_L \end{bmatrix} \prec 0,P_1 \succ 0, \dots, P_L \succ 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/50fc71737b69f2cf57d15634f2f19d091bf37d02

Conclusion
The stability of the above linear delayed differential equation is proved, using Lyapunov functionals of the form $$ V(x,t)=x(t)^\top P x(t)+\sum_{i=1}^{L}\int_{0}^L x^\top(t-s)P_i x(t-s) \ ds $$, if the provided LMIP is feasible.

Remark
The techniques for proving stability of norm-bound LDIs [Boyd, ch.5] can also be used.