Obtaining all possible transitions based on the estimation of a subset of transitions through the linearity property in an optimal control problem

Question

I found an article in which the authors formulated an optimal control problem based on a linear time-invariant system

$$ \dot{x}(t)=Ax(t)+Bu(t) $$

with initial state $ x(0)=x_0 $ and final state $ x(T)=x_T $.

The performance measure to be minimized was:

$$ \int_{0}^{T} ((x_T-x(t))^T(x_T-x(t))+\rho u(t)^Tu(t) dt $$ with $ \rho=100 $ and $ T=1$

The authors studied a set of 56 transitions, where the initial states and final states were represented by a binary vector of length 129 respectively.

Now the authors note the following:

Importantly, because our dynamical model is linear, any possible transition can be written as a linear combination of these transitions (though the resulting transition may not be optimal, in terms of minimum energy). Thus, our results are generally relevant to all transitions.

Can somebody help me to understand this or link some information on this? In other words, after the estimation of a set of transitions based on the minimization of some performance measure, how to calculate all other transitions as a linear combination of the estimated set of transitions?

Link to article: https://www.nature.com/articles/srep30770

Answer 1

If one would solve this problem for example with PMP then you would get the following state and co-state dynamics

$$ \begin{bmatrix} \dot{x} \\ \dot{\lambda} \end{bmatrix} = \underbrace{ \begin{bmatrix} A & -\frac{1}{2\,\rho}B\,B^\top \\ -2\,I & -A^\top \end{bmatrix}}_\hat{A} \begin{bmatrix} x \\ \lambda \end{bmatrix} + \underbrace{ \begin{bmatrix} 0 \\ 2\,I \end{bmatrix}}_\hat{B} x_T $$

In general those dynamics have to the following solution in time

$$ \begin{bmatrix} x(t) \\ \lambda(t) \end{bmatrix} = e^{\hat{A}\,t} \begin{bmatrix} x(0) \\ \lambda(0) \end{bmatrix} + \int_0^t e^{\hat{A}(t - \tau)}d\tau\,\hat{B}\,x_T. $$

By defining the following submatrices

$$ e^{\hat{A}\,T} = \begin{bmatrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{bmatrix}, \quad \int_0^T e^{\hat{A}(T - \tau)}d\tau\,\hat{B} = \begin{bmatrix} Y_1 \\ Y_2 \end{bmatrix} $$

then it is possible obtain the following expression

$$ x(T) = X_{11}\,x(0) + X_{12}\,\lambda(0) + Y_1\,x_T $$

since $x(T)=x_T$ and $x(0)=x_0$, then this can be used to solve for $\lambda(0)$

$$ \lambda(0) = X_{12}^{-1} ((I - Y_1)\,x_T - X_{11}\,x_0) = \begin{bmatrix} -X_{12}^{-1}\,X_{11} & X_{12}^{-1} (I - Y_1) \end{bmatrix} \begin{bmatrix} x_0 \\ x_T \end{bmatrix}. $$

Combining this with the expression for the state as a function of time yields

$$ \begin{bmatrix} x(t) \\ \lambda(t) \end{bmatrix} = e^{\hat{A}\,t} \begin{bmatrix} I & 0 \\ -X_{12}^{-1}\,X_{11} & X_{12}^{-1} (I - Y_1) \end{bmatrix} \begin{bmatrix} x_0 \\ x_T \end{bmatrix} + \int_0^t e^{\hat{A}(t - \tau)}d\tau\,\hat{B}\,x_T $$

so from this it can be concluded that the solution of the state $x$ as a function of time should be linear in $x_0$ and $x_T$

$$ x(t) = \Phi(t) \begin{bmatrix} x_0 \\ x_T \end{bmatrix} $$

where $\Phi(t)$ is a function of time, independently of $x$. But the solution for $\Phi(t)$ might change if a difference $A$, $B$, $\rho$ or $T$ would be used. If $x\in\mathbb{R}^n$ then $\Phi(t)\in\mathbb{R}^{n\times2\,n}$.

I have only quickly skimmed through the paper you linked to, but I am not sure why that author states there are only 56 "transitions" (which should correspond to the columns of $\Phi(t)$), while the state space dimension is of size 129. Since in the general case I would actually expect twice as many transitions as states, so 258. But maybe the author put constraints on the allowed $x_0$ and $x_T$ which reduces the span of $\Phi(t)$.