Representations of mild solutions of time-varying linear stochastic equations and the exponential stability of periodic systems. (English) Zbl 1072.60047

Ungureanu, Viorica Mariela

Electron. J. Qual. Theory Differ. Equ. 2004, Paper No. 4, 22 p. (2004).

Chapter 1 contains preliminaries. Let \((\Omega, F,{\mathcal F}_t\), \(t\in [0,\infty), P)\) be a stochastic basis and \(L^2_s(H)= L^2(\Omega,{\mathcal F}_s, P,H)\) and consider the stochastic equation \[ dy(t)= A(t) y(t)\,dt+ \sum^m_{i=1} G_i(t) y(t)\,d\omega_i(t).\tag{1} \] The author associates to (1) the approximating system \[ dy_n(t)= A_m(t) y_n(t)\,dt+ \sum^m_{i=1} G_i(t) y_m(t)\,d\omega_i(t). \] Also, she considers the Lyapunov equation \[ {dQ(s)\over ds}+ A^*(s) Q(s)+ Q(s)A(s)+ \sum^m_{i=1} G^*_i(s) Q(s) G_i(s)= 0,\quad s\geq 0.\tag{2} \] \(Q(s)\) [see G. Da Prato and A. Ichikawa, Syst. Control Lett. 9, 165–172 (1987; Zbl 0678.93051)] is a mild solution on the interval \(J\subset\mathbb{R}\) of (2).
Chapter 2 treats “Differential equations on \({\mathcal H}_2\)”. Chapter 3 is entitled “The covariance operator of the mild solutions of linear stochastic differential equations and the Lyapunov equations”. This chapter contains the proof of the theorem: Let \(V\) be another real separable Hilbert space and \(B\in L(H, V)\). If \(y(t,s;\xi)\), \(\xi\in L^2_s(H)\), is the mild solution of (2) and \(Q(t,s,R)\) is the unique mild solution of (2) with the final value \(Q(t)= R\geq 0\), then
a) \((E[y(t, s;\xi)\otimes y(t,s,\xi)])= \text{Tr\,}Q(t,s,u\otimes u)E(\xi\otimes\xi)\) for all \(u\in H\);
b) \(E\| By(t,s;\xi)\|= \text{Tr\,}Q(t,s,B^* B)E(\xi\otimes\xi)\).
Chapter 4 is devoted to “The solution operators associated to the Lyapunov equations”. The next chapter “The uniform exponential stability of linear stochastic systems with periodic coefficients” contains the proof of the theorem: The following assertions are equivalent:
a) equation (1) is uniformly exponential stable;
b) \(\lim_{n\to\infty} E\| y(n,\tau,0; x)\|^2= 0\) uniformly for \(x\in H\), \(\| x\|= 1\);
c) \(\rho(T(\tau, 0))< 1\).

Reviewer: G. G. Vrănceanu (Bucureşti)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

stochastic differential equation; Lyapunov equations; uniform exponential stability

Citations:

Zbl 0678.93051