##
**Delay-dependent robust stability and \(H_\infty\) control of jump linear systems with time-delay.**
*(English)*
Zbl 1015.93069

The authors consider stability properties of a stochastic control system \(\Sigma_1\) that is linear in the state with random coefficients and delay \(\tau\), given by the dynamics
\[
\dot{x}_t = A(r_t)x_t + A_1(r_t)x_{t-\tau} + B(r_t)u_t + B_w(r_t)w_t,
\]
the initial condition \(x_t = \Phi(t)\), and \(r_t = r_0\) for \(t \in [-\tau, 0], (\tau>0)\) and the output
\[
z_t = C(r_t)x_t + D(r_t)u_t + E(r_t)w_t,
\]
where \(u_t\) is a control, \(w_t\) is a noise process and \(r_t\) a finite state Markov process with values in \(S = \{1,2,\dots ,N\}\) and \(A, A_1,\dots ,E\) are matrix-valued functions of polytype \({\mathcal M}(i),\) meaning that they belong to a family of matrix-valued functions \(M\) on \(S\) with the property that there is a number \(\nu \in {\mathbb N}\) and non-negative coefficients \(t_j\) \((j=1,\dots ,\nu)\) with \(\sum_{j=1}^{\nu}t_{j} = 1\) that are independent of \(M\) such that any value \(M(i)\) can be represented as the convex combination \(M(i) = \sum_{j=1}^{\nu}t_jM_{ij}\) of suitable matrices \(M_{ij}\).

Sufficient conditions are given in terms of algebraic inequalities for the involved coefficient matrices and the delay \(\tau\) (known as linear matrix inequalities) for (i) the uncontrolled system \(\Sigma_1\) with \(u_t=0\) and (ii) the closed-loop feedback system \(\Sigma_1\) with \(u_t=K(r_t)x_t\) to be robustly stochastically stable or, respetively, robustly stable with noise attenuation level \(\gamma\) (\(\gamma>0\) given) that is satisfying \( \int_{0}^{\infty}{\mathbb E}{\mathbf [} \|x_t\|^2 \mid r_0,\Phi(\cdot){\mathbf ]} dt \leq M_0(r_0, \Phi(\cdot), \|w\|_2) \) or \( {\mathbb E} \{ \int_{0}^{\infty} \|z_t\|^2 dt \}^{\frac{1}{2}} \leq \gamma [M(r_0, \Phi(\cdot)) + \|w\|_2] \), respectively, with suitable constants \(M_0\) and \(M\) (depending only on the initial conditions [and noise \(w\)]) satisfying \(M_0(r_0,0,0) = M(r_0,0) = 0 \).

If the system \(\Sigma_1\) is modified to a system \(\Sigma_3\) with delay also in the control, an algorithm is given for constructing a feedback control such that \(\Sigma_3\) with that control is robustly stable for all delays \(\tau\) that vary over a given interval.

Sufficient conditions are given in terms of algebraic inequalities for the involved coefficient matrices and the delay \(\tau\) (known as linear matrix inequalities) for (i) the uncontrolled system \(\Sigma_1\) with \(u_t=0\) and (ii) the closed-loop feedback system \(\Sigma_1\) with \(u_t=K(r_t)x_t\) to be robustly stochastically stable or, respetively, robustly stable with noise attenuation level \(\gamma\) (\(\gamma>0\) given) that is satisfying \( \int_{0}^{\infty}{\mathbb E}{\mathbf [} \|x_t\|^2 \mid r_0,\Phi(\cdot){\mathbf ]} dt \leq M_0(r_0, \Phi(\cdot), \|w\|_2) \) or \( {\mathbb E} \{ \int_{0}^{\infty} \|z_t\|^2 dt \}^{\frac{1}{2}} \leq \gamma [M(r_0, \Phi(\cdot)) + \|w\|_2] \), respectively, with suitable constants \(M_0\) and \(M\) (depending only on the initial conditions [and noise \(w\)]) satisfying \(M_0(r_0,0,0) = M(r_0,0) = 0 \).

If the system \(\Sigma_1\) is modified to a system \(\Sigma_3\) with delay also in the control, an algorithm is given for constructing a feedback control such that \(\Sigma_3\) with that control is robustly stable for all delays \(\tau\) that vary over a given interval.

Reviewer: Volker Wihstutz (Charlotte)

### MSC:

93E15 | Stochastic stability in control theory |

93D09 | Robust stability |

93C23 | Control/observation systems governed by functional-differential equations |

93B36 | \(H^\infty\)-control |

93E20 | Optimal stochastic control |

34K50 | Stochastic functional-differential equations |