Pontryagin's maximum principle for the Roesser model with a fractional Caputo derivative (Q6577202)

The system is\N\begin{align*}\ND^{\alpha_1}_1 z_1(x) &= a_1(x)z_1(x) + a_2(x)z_2(x) + f_1(x, u(x)) \\\ND^{\alpha_2}_2 z_2(x) &= b_1(x)z_1(x) + b_2(x)z_2(x) + f_2(x, u(x))\N\end{align*}\Nwith \(x = (x_1, x_2),\) \(0 <\alpha_1, \alpha_2 < 1\) and \(D^{\alpha_1}_1, D^{\alpha_2}_2\) are the Caputo derivatives\N\begin{align*}\N\big(D_1^{\alpha_1}w\big)(x_1, x_2) &= \frac{1}{\Gamma(1 - \alpha_1)}\frac{\partial}{ \partial x_1} \int_{x^0_1} ^{x_1}(x_1 - s_1)^{- \alpha_1}\big(w(s_1, x_2) - w(x^0_1, x_2)\big)ds_1 \\\N\big(D_2^{\alpha_2}w\big)(x_1, x_2) &= \frac{1}{\Gamma(1 - \alpha_2)}\frac{\partial}{ \partial x_2} \int_{x^0_2} ^{x_2}(x_2 - s_2)^{- \alpha_2}\big(w(x_1, s_2) - w(x_1, x_2^0)\big)ds_2 \, .\N\end{align*}\NThe initial conditions are\N\[\Nz_1 \big( x^0_1, x_2\big) = \varphi_{10}(x_2) \, , \quad z_2 \big( x_1, x_2^0\big) = \varphi_{01}(x_1)\N\]\Nand the goal is to minimize the integral\N\begin{align*}\NJ(u) &= \frac{1}{ \Gamma(\beta_1)} \frac{1}{\Gamma(\beta_2)} \int_{x^0_1}^{X_1} \int_{x^0_2}^{X_2}\big(X_1 - x_1\big)^{\beta_1 - 1} (X_2 - x_2)^{\beta_2 - 1} \\\N& \ \ \ \times \big(c'_1(x)z_1(x) + c'_2(x)z_2(x) + f_0(x, u(x)) \big) dx \\\N&\ \ \ + \frac{1}{\Gamma(\beta_1)} \int_{x^0_1} ^{X_1} (X_1 - x_1)^{\beta_1 - 1} d'(x_1) z_2(x_1, X_2) dx_1 \\\N& \ \ \ + \frac{1}{\Gamma(\beta_2)} \int_{x^0_2} ^{X_2} (X_2 - x_2)^{\beta_2 - 1} d'(x_2) z_1(X_1, x_2) dx_2\N\end{align*}\Nwith a control constraint \(u(x) \in V.\) Under suitable assumptions, the authors prove an existence theorem for this optimal problem and an optimality condition in the form of a maximum principle\N\[\NH(x, \psi(x), u(x)) = \max_{v \in V}H(x, \psi(x), v) \, .\N\]\NThe results are illustrated with a numerical example. The authors remark that their results can also be used to construct approximate solutions.