Preamble

This paper considers a dynamical system described by the following functional differential equation with a state-dependent delay:
$$ \frac{dx(t)}{dt} = f[x(t), x(t - h(x, t)), u(t), t], \quad t \in T = [t_0, t_1] $$
subject to the initial conditions $x(\tau) = \phi(\tau)$ for $\tau \in S_0$, where $S_0$ is the initial set. Here, $x = (x_1, \dots, x_n)$ is the state vector in $E^n$, $u = (u_1, \dots, u_r)$ is the control vector belonging to a compact set $U \subset E^r$, and $h(x, t)$ is the delay function.

We assume the following conditions hold:
1. The function $f(x, y, u, t)$ is defined on the domain $P = X \times X \times U \times T$ and satisfies a Lipschitz condition:
$$|f(x', y', u, t) - f(x, y, u, t)| \leq L_1 (|x' - x| + |y' - y|)$$
2. The delay function $h(x, t)$ is defined on $Q = X \times T$ and satisfies:
$$|h(x', t) - h(x, t)| \leq L_2 |x' - x|$$
3. The initial function $\phi(t)$ is defined on $S_0$ and is Lipschitz continuous:
$$|\phi(t') - \phi(t)| \leq L_3 |t' - t|$$
4. The function $f$ is bounded such that $|f(x, y, u, t)| \leq L_4$ for all points in $P$.

1. Existence and Uniqueness of Solutions

To prove the existence of a solution $x(t)$ on the interval $[t_0, t^]$, we employ the method of successive approximations. Let $x_0(t) = \phi(t_0)$ for $t \in [t_0, t^]$ and define the sequence:
$$ x_{m+1}(t) = \phi(t_0) + \int_{t_0}^t f(x_m(\tau), x_m(\tau - h(x_m(\tau), \tau)), u(\tau), \tau) d\tau $$
where $x_m(\tau) = \phi(\tau)$ if $\tau \in S_0$. For a sufficiently small interval $[t_0, t^]$, where $t^ = t_0 + \Delta$, the sequence ${x_m(t)}$ converges uniformly to a unique solution $x(t)$. The error estimate for the $m$-th approximation is given by:
$$ |x(t) - x_m(t)| \leq 2b L^m (2 + L_2)^{m-1} \frac{(t - t_0)^m}{m!} $$
where $b$ is a constant related to the initial bounds of the function.

2. Stability and Continuous Dependence

We further analyze the dependence of the solution on the control $u(t)$ and the initial function $\phi(t)$. If $x(t)$ and $\bar{x}(t)$ are solutions corresponding to controls $u(t)$ and $\bar{u}(t)$, then for any $\epsilon > 0$, there exists a $\delta > 0$ such that if $\int_{t_0}^{t_1} |u(\tau) - \bar{u}(\tau)| d\tau < \delta$, then $|x(t) - \bar{x}(t)| < \epsilon$ for all $t \in T$.

Specifically, the difference between solutions can be bounded as:
$$ |x(t) - \bar{x}(t)| \leq (M_1 + M L_2) \delta e^{L_1(2 + M L_2)(t - t_0)} $$
where $M = \max{L_3, L_4}$. This demonstrates that the system is stable with respect to small perturbations in the control input and initial data.

3. Optimal Control and Differential Inclusions

Consider the problem of minimizing a functional $J(u) = \Phi(x(t_1))$. To analyze the set of reachable states, we define the differential inclusion:
$$ \frac{dx(t)}{dt} \in R(x(\cdot), t) $$
where $R(x(\cdot), t) = { z : z = f[x(t), x(t - h(x(t), t)), u, t], u \in U }$.

The set of trajectories $X_t(\cdot)$ is compact in the space of continuous functions. For any trajectory $z(t)$ of the relaxed system (the convex hull of the velocity set), there exists a sequence of trajectories of the original system (1) that converges uniformly to $z(t)$. This allows us to apply the Filippov-type existence theorems for optimal control.

4. Necessary Conditions for Optimality

Using the properties of the reachable set and the continuity of the mapping $R(x(\cdot), t)$, we can derive necessary conditions for the optimal control $u(t)$. If $u^*(t)$ is an optimal control, then there exists a non-zero adjoint vector function $\psi(t)$ satisfying the corresponding adjoint system, such that the Hamiltonian reaches its maximum:
$$ H(x, \psi, u, t) = \max_{u \in U} (\psi(t), f[x, y, u, t]) $$
The presence of the state-dependent delay $h(x, t)$ introduces additional terms into the adjoint equation involving the derivative of the delay with respect to the state $x$.

References

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