Lyapunov’s Stability Theorem

Let $x = 0$ be an equilibrium point for a dynamic system $\dot{x}=f(x)$ and $D \subset R^n$ be a domain containing $x=0$. Let $V: D \to R$ be a continuously differentiable function such that

\[V(0)=0 \quad and \quad V(x)>0 \quad in \quad D-\{0\}, \quad \dot{V}(x) \leq 0 \quad in \quad D\]

Then $x = 0$ is Lyapunov stable. Moreover, if $\dot{V}(x) < 0$ in $D$, then $x=0$ is asymptotically stable.

Sensitivity Constraints

\[\int_{0}^{\infty}\ln|S(j\omega)|d\omega=\pi\sum_{i=1}^{N_p}R(p_i) > 0\]

Lagrange Multipliers


Optimal Control (HJB)

\[0=J^*_t(x(t),t)+\min_{u(t)\in U}\{g(x(t),u(t),t)+J^*_t(x(t),t)a(x(t),u(t),t)\}\]

Riccati Equation


Bayes’ Rule


Least Squares


where $A^{\dagger}$ is the pseudoinverse or Moore–Penrose generalized inverse of A.

Small Gain

\[||L(j\omega)|| < 1 \quad for \quad all \quad \omega\]




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