Multiple solutions in optimal control

Consider a control problem of the form:

$\frac{d \vec{x}}{dt} = F(\vec{x}, \vec{u})$

where $\vec{u}$ is the control inputs and $\vec{x}$ is the state variables, and all we want to do is drive the system from $\vec{x}_0$ to $\vec{x}_1$ in some given amount of time $T$.

I have two questions:

1. What is known about how many solutions for $\vec{u}$ exist for such a problem?
2. How do engineers address/interpret the existence of multiple solutions?
• Do you have a particular cost function associated with the dynamics and the parameters w? This will answer a lot about how many solutions exist. Even if multiple solutions exist, many solutions might not be desirable. In which case, you will want to add a regularization term to your cost function which biases your toward a solution with desirable qualities. – Paul Feb 26 '16 at 6:20

2 Answers

I assume that you also want to minimize some function of $t$, $T$, $\vec{x}(t)$ and $w(t)$. In this case it depends on how you define a solution to this problem. If you define a solution as minimization of that function, then you will have multiple solutions if there are multiple local minima, but the global minima would be the best solution. If there are multiple solutions with the same functional value as the global minima, then that would just mean that there are multiple "trajectories" from $\vec{x}_0$ to $\vec{x}_1$ which are equally good. You would have to choose one of them or adjust your cost function.

Optimal Control Theory

I will talk about the Hamilton-Jacobi-Bellman equation. But, for more information, you can read this book. There are information about Dynamic Programming and Calculus of Variation for optimal control problems.

Suppose you have a system with the dynamic described by the state equation:

$$\dot{\vec{x}} = f(\vec{x}, \vec{u})$$

were $$\vec{x}$$ is the state variables and $$\vec{u}$$ the control inputs.

Usually, to control a system, we wish to minimize a performance measure:

$$J = h(\vec{x}(T), \vec{u}(T)) + \int_{0}^{T} g(\vec{x}(t), \vec{u}(t), t) dt$$

were h($$\cdot$$) and g($$\cdot$$) are known as the terminal cost and functional, respectively.

For your first question:

What is known about how many solutions for $$\vec{u}$$ exist for such a problem?

The number of feasible control inputs, $$\vec{u}$$, are not relevant for the problem. The question we are looking for the answer is: there exist an optimal control law, $$\vec{u}^{*}$$, such that the performance measure is satisfied?

For your second question:

How do engineers address/interpret the existence of multiple solutions?

To deal with those feasible solutions, we try to solve the minimization problem.

Defining the Hamiltonian as:

$$\mathcal{H} = g(\vec{x}(t), \vec{u}(t), t) + \frac{\partial J}{\partial x}\big( f(\vec{x}, \vec{u}) \big)$$

Is necessary and, for no boundaries conditions, sufficient to show that:

$$\frac{\partial\mathcal{H}}{\partial \vec{u}} = 0$$ and $$\frac{\partial^{2}\mathcal{H}}{\partial \vec{u}^{2}} > 0$$

The solution of this partial derivative of the Hamiltonian will give you the optimal control law, $$\vec{u}^{*}$$.

I hope this is useful. Sorry if it is a little to much abstract.

Best regards.