关于分布参数系统最佳调节器的稳定裕度

• 作 者: 李训经
• 机 构: 复旦大学
• 出 处: 控制理论与应用
• 发表日期: 1986年01期
• 关键词: 强连续 半群 稳定裕度 分布参数系统 调节器问题 线性有界算子 多变量系统 基本引理 世界大会 扩稳

本文直接证明 定理 假设 1.x(·)是dx(t)/dt=Ax(t)+Bu(t),x(0)=x_0的mild解,其中A是Hilbert空间X上的强连续半群e~(At)(t≥0)的母元,u(·)∈L~2([0,+∞),Z), 2.当integral from n=0 to ∞ (‖u(t)‖~2dt<+∞)和integral from n=0 to ∞ (‖Lx(t)‖~2dt<+∞)时integral from n=0 to +∞ (‖x(t)‖~2dt<+∞) ; 3.P≥0满足A~TP+PA+L~TL-PBR~(-1)B~TP=0和integral from n=0 to +∞ (‖e~((A-BR~(-1)B~TP)t)‖~2dt<+∞,其中R(?)0; 4.n(·): Z→Z是强连续的,且存在k>0和β>0使得integral from n=0 to +∞ (‖n(n(t))‖~2dt≤k integral from n=0 to +∞ (‖u(t)‖~2dt,(1+β)/2)integral from n=0 to +∞ dt≤integral from n=0 to +∞ dt. 那末方程的mild解是渐近稳定的。 This paper directly proves that the theorem 1.x (·) is a mild solution of dx (t) / dt = Ax (t) + Bu (t), x (0) = x_0, where A is a strong continuous half on Hilbert space X U (·) ∈ L ~ 2 ([0, + ∞), Z) for the group e ~ (At) (t≥0); 2. When the integral from n = 0 to ∞ (‖u (t) (‖x 2t + ∞) and integral from n = 0 to ∞ (‖Lx (t) ∥ ~ 2dt <+ ∞) ; 3.P ≧ 0 satisfies A ~ TP + PA + L ~ TL-PBR ~ -1 B ~ TP = 0 and integral from n = 0 to + ∞ (A ~ BR ~ () 0; 4.n (·): Z → Z is strongly continuous, and the existence of k> 0 and β> 0 makes integral from n = 0 to + ∞ (‖n (n (t)) ‖ ~ 2dt ≦ k integral from n = 0 to + ∞ (‖u (t) ∥ ~ 2dt, (1+ β) / 2) integral from n = 0 to Then the mild solution of the equation is asymptotically stable. If ∞ dt≤integral from n = 0 to + ∞