How can I define uniform convergence for this sequence of functions?

Question

Consider the following recurrence relation $$ x_{k+1} = Ax_k + Bu_k $$ where

• $x_0 \in \mathbb R^n$ is the initial condition,
• $A \in \mathbb R^{n \times n}$ and $B \in \mathbb R^{n \times m}$, and
• $u_k \in \mathbb R^m$ is an external input at the $k$th timestep (can be chosen as desired).

Suppose we then define the following "cost" function $$ J_{N-1}\left(x_0,u_0,\dots,u_{N-1}\right) = \sum_{k=0}^{N-1} x_k^T Q x_k + u_k^T R u_k \tag{1} $$ where $Q \in \mathbb R^{n \times n}$ is positive semi-definite and $R \in \mathbb R^{m \times m}$ is positive definite. Additionally, let $\mathcal U(x_0)$ be the set of infinite sequences of vectors $(u_0,u_1,\dots)$ such that, for the given initial condition $x_0$,

• the chosen sequence $(u_0,u_1,\dots) \in \mathcal U(x_0)$ results in $\lim_{k \to \infty} x_k = 0$, and
• $\exists M \in \mathbb N, k \geq M \implies u_k = 0$.

I want to show that, given the initial condition $x_0$, when $(u_0,u_1,\dots) \in \mathcal U(x_0)$, the objective function in $(1)$ converges uniformly to some value $J_\infty(x_0,u_0,u_1,\dots)$ that depends on $x_0$ and $(u_0,u_1,\dots)$.

However, I'm having trouble defining uniform convergence in this case, since each of the functions in the sequence $J_0(x_0,u_0),J_1(x_0,u_0,u_1),J_2(x_0,u_0,u_1,u_2),\dots$ have different domains. Any suggestions on how I can define uniform convergence here?

Answer 1

The cost should be here a function of $x_0$ only here as the sequence of inputs is chosen a priori. As a result, we define

$$ J_{N-1}(x_0) = \sum_{k=0}^{N-1}\left(x_k^T Q x_k + u_k^T R u_k\right). $$

For the sum to converge you need that $\sum_{k=0}^{N-1}x_k^T Q x_k$ to converge to a constant, so you need something stronger than just $x_k\to0$. The issue is that the input becomes zero after a certain time, so you will

• either need $\{Q^{1/2}y_k\}_{k\in\mathbb{N}}\in\ell_2$ where $y_{k+1}=Ay_k$ (which is weaker than asking for $A$ to be Schur stable), or
• need the input to drive the state to a value $x(M)$ for which $Q^{1/2}x(M)=0$ and $Ax(M)=x(M)$. This includes $x(M)=0$ as special case.

With any of those assumptions, you will directly get the pointwise convergence of $J_N(x_0)$ to $J_\infty(x_0)$. The uniform convergence (i.e. for all $x_0$, possibly in some compact set) is more difficult to prove and you will certainly need more assumptions.