## 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.

### 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
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