Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations

B

Proof of Theorem 8.2.2

We begin with a preliminary lemma.

Consider the nonlinear time-varying system:

$\dot{x}\left(t\right)=\varphi \left(x\left(t\right),w\left(t\right)\right),x\left(0\right)=x_0$

(B.1)

where x(t) ∈ℜⁿ is the state of the system and w(t) ∈ℜ^r is an input, together with the cost functional:

$J_i\left(x(.),w(.)\right)={\displaystyle {\int }_0^{\infty }{\mu }_i\left(x\left(t\right),w\left(t\right)\right)dt,i=0,\mathrm{\dots },l.}$

(B.2)

Suppose the following assumptions hold:

(a1) ϕ(., .) is continuous with respect to both arguments;

(a2) For all {x(t), w(t)} ∈ L₂[0, ∞), the integrals (B.2) above are bounded:

(a3) For any given ∈ > 0, w ∈ W, there exists a δ > 0 such that for all x₀ ≤ δ, it implies:

$\left|J_i\left(x_1\left(t\right),w\left(t\right)\right)-J_i\left(x_2\left(t\right),w\left(t\right)\right)\right|<\in ,i=0,\mathrm{\dots },l,$

where x₁(.), x₂(.), are any two trajectories of the system corresponding to the initial conditions x₀ = 0.

We then have the following lemma ...