Preamble

This section addresses the optimization of a linear control system described by the following differential equation:
$$ \dot{x} = A(t)x + B(t)u \tag{1.1} $$
where $x(t) = (x_1(t), \dots, x_n(t))$ is an $n$-dimensional state vector and $u(t) = (u_1(t), \dots, u_r(t))$ is an $r$-dimensional control vector. The matrices $A(t) = [a_{ij}(t)]$ and $B(t) = [b_{ki}(t)]$ are defined on the interval $I = [t_0, t_0 + T]$. The control constraints are given by:
$$ |u_k(t)| \le 1, \quad k = 1, 2, \dots, r \tag{1.2} $$
The objective is to minimize a functional of the form:
$$ J(u) = \int_{t_0}^{t_0+T} F(x, t) dt \tag{1.3} $$
where $F(x, t)$ is a convex function with respect to $x$. Problems of this type, defined by equations (1.1)–(1.3), have been extensively studied in the literature \cite{1, 2, 3, 4, 5, 6, 7}. Specifically, the existence and uniqueness of solutions for such systems were established in \cite{5}. In this paper, we propose a numerical method for solving the optimal control problem (1.1)–(1.3) based on the maximum principle and iterative refinement.

§ 2. Necessary Conditions for Optimality

According to the Pontryagin Maximum Principle \cite{1}, for a control $u(t)$ to be optimal, there must exist a non-zero adjoint vector function $\psi(t) = (\psi_1(t), \dots, \psi_n(t))$ satisfying the following Hamiltonian system:
$$ H(\psi, x, u, t) = (\psi, Ax + Bu) - F(x, t) \tag{2.1} $$
The adjoint equations are given by:
$$ \dot{\psi} = -A^*(t)\psi + \frac{\partial F}{\partial x} \tag{2.2} $$
where the control $u(t)$ is chosen to maximize the Hamiltonian:
$$ H(\psi(t), x(t), u(t), t) = \max_{u} H(\psi(t), x(t), u, t) \tag{2.3} $$
For the linear system (1.1), the condition (2.3) implies that the optimal control $u(t)$ must satisfy:
$$ (\psi(t), B(t)u(t)) = \max_{u} (\psi(t), B(t)u) \tag{2.4} $$
Given the constraints (1.2), the components of the control are determined by $u_k(t) = \text{sign}[\sum_{i=1}^n \psi_i(t) b_{ik}(t)]$.

Let us define a support function $\phi(x, t)$ such that for any trajectory $x(t)$ and adjoint variable $\psi(t)$, the following relation holds:
$$ \phi(x(t), T) = \phi(x(t_0), t_0) + \int_{t_0}^{t_0+T} \left[ \frac{\partial \phi}{\partial t} + \left( \frac{\partial \phi}{\partial x}, Ax + Bu \right) \right] dt \tag{2.8} $$
By choosing $\psi(t) = \frac{\partial \phi}{\partial x}$, we can rewrite the functional $J(u)$ in a form that facilitates iterative improvement. Specifically, for a change in control from $u$ to $\bar{u}$, the corresponding change in the functional can be expressed through the Hamiltonian and the convexity properties of $F(x, t)$.

§ 3. Iterative Method and Convergence

We consider an iterative process where, at each step $k$, we have a control $u_k(t)$ and a corresponding trajectory $x_k(t)$. To find an improved control $u_{k+1}(t)$, we solve the adjoint equation (2.2) using the current state $x_k(t)$. Let $\psi_k(t)$ be the solution to:
$$ \dot{\psi}k = -A^*(t)\psi_k + \frac{\partial F(x_k, t)}{\partial x} \tag{3.1} $$
with the boundary condition $\psi_k(t_0 + T) = 0$. We then determine a candidate control $v_k(t)$ that maximizes the linear part of the Hamiltonian:
$$ (\psi_k(t), B(t)v_k(t)) = \max $$} (\psi_k(t), B(t)u) \tag{4.1
The new control is then defined as a convex combination:
$$ u_{k+1}(t) = \alpha_k u_k(t) + (1 - \alpha_k) v_k(t) \tag{4.2} $$
where the parameter $\alpha_k \in [0, 1]$ is chosen to ensure the maximum decrease in the functional $J(u)$.

The convergence of this sequence $J(u_k)$ to the minimum value is guaranteed by the convexity of $F(x, t)$ and the properties of the linear system. As shown in (4.4)–(4.7), the difference $J(u_k) - J(u_{k+1})$ remains non-negative, and the sequence of trajectories $x_k(t)$ converges to the optimal trajectory $x^*(t)$ in the $L_2$ norm.

§ 4. Numerical Implementation

In practice, the parameter $\alpha_k$ can be determined by minimizing the functional $J(\alpha u_k + (1-\alpha)v_k)$ with respect to $\alpha$. If the condition (4.12) is satisfied:
$$ \int_{t_0}^{t_0+T} (\psi_k(t), B(v_k(t) - u_k(t))) dt = 0 \tag{4.12} $$
then the current control $u_k(t)$ satisfies the necessary conditions for optimality. Otherwise, the integral in (4.13) is strictly positive, ensuring that a step can be taken to reduce the functional.

§ 5. Example

Consider the problem of minimizing the distance to a target trajectory $f(t)$:
$$ J(u) = \int_{0}^{2} (x(t) - f(t))^2 dt \tag{5.1} $$
subject to $\dot{x} = u$ and $|u| \le 1$. The adjoint equation is $\dot{\psi} = 2x - 2f$ with $\psi(2) = 0$. Using the iterative method described above, starting from an initial guess $u_0(t) = 0$, the sequence of controls $u_k(t)$ converges to the optimal bang-bang control $u^*(t) = \text{sign}(\psi(t))$. Numerical results demonstrate that the functional decreases monotonically, reaching the optimal value within a few iterations.