Differential Equation (Prove)

Question

Given the control system

d/dt [x₁(t)/x₂(t)] = [-x₁(t)/(x₂(t)].u(t)

where u(t)∈[-1,1] and x(t)∈R² for all t ∈ [0,1], and x(0)=[x₁(0),x₂(0)]^T=: x⁰.

1. Prove there is an optimal controller for every x0 and there exists

min IIx(1)II₂²

u:[0,1]→[-1,1]

1. For every x0 ∈ R2, find an optimal controller u:[0,1]→[-1,1]

Answer 1

An optimal control exists for every initial state x⁰ ∈ R²

The problem is convex and well-posed

The obtained solution yields the global minimum