Limit cycles for two classes of control piecewise linear differential systems

We study the bifurcation of limit cycles from the periodic orbits of $2n$–dimensional linear centers $\dot{x}=A_{0}x$ when they are perturbed inside classes of continuous and discontinuous piecewise linear differential systems of control theory of the form $\dot{x} = A_0 x + \varepsilon \big(A x + \phi(x_1) b\big)$,where$\phi$ is a continuous or discontinuous piecewise linear function, $A_0$ is a $2n\times 2n$ matrix with only purely imaginary eigenvalues, $\epsilon$ is a small parameter, $A$ is an arbitrary $2n\times 2n$matrix,and$b$isanarbitraryvectorof$$\mathbb{R}$^n$.