Question

Use LaSalle's theorem to show that the origin is asymptotically stable for the following systems: i. Consider the linear system x =(A−BR −1 B T P)x, where P=P T ≻0 satisfies the Riccati equation PA+A TP+Q−PBR −1B T P=0 R=R T ≻0, and Q=Q T ≽0. Using V(x)=x T

Px as a Lyapunov function candidate, when Q=C T C and (A,C) is observable. ii. x˙ =x 2 ,˙2 =−x 1 − 1+x 2/2x /2
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Answer 1

Using LaSalle's theorem and the provided Riccati equation, we demonstrated that for the linear system with a given Lyapunov function candidate, the origin is asymptotically stable since the derivative of the Lyapunov function is nonpositive, and zero only at the origin.

Step-by-step explanation:

To show that the origin is asymptotically stable for the given linear system using LaSalle's theorem, we first note the provided system equation:

x = (A-BR-1 BT P)x,

where P = PT ≫ 0 satisfies the Riccati equation PA + ATP + Q - PBR-1BTP = 0, R = RT ≫ 0, and Q = QT ≥ 0. Using V(x) = xTPx as a Lyapunov function candidate, where Q = CTC and the pair (A,C) is observable, we need to demonstrate that the derivative of V along trajectories of the system is nonpositive and the only points where the derivative of V is zero are on the trajectory x=0.

Computing the derivative of V along the trajectories gives:

∂V/∂t = (Ax + BR-1BTPx)TPx + xTP(Ax + BR-1BTPx)

= xT(ATP + PA)x + xTPBR-1BTPx + xTPBR-1BTPx

Using the Riccati equation to substitute ATP + PA, we get:

= -xTQx

Since Q = CTC, and C is the output matrix in a controllable system, the only solution for V to be zero is x=0. Hence, by LaSalle's theorem, the origin is asymptotically stable.

The second system given is non-linear and contains a typographical error. It appears this part of the question is not formulated correctly and thus cannot be addressed as presented.