Boundary sliding mode control of a system of linear hyperbolic equations: A Lyapunov approach

In the input quantization case, the locally Lipschitz assumption on the quantizer is not required because the function subject to quantization can be treated as a finite-dimensional object (the exact predictor state), and well-posedness can be studied by combining results for ODE systems with [26].

“Nevertheless, in the input quantization case such an assumption is not required as the function that is subject to quantization can be treated as a finite-dimensional object (in fact, it is the exact predictor state), and thus, well-posedness can be studied combining results for ODE systems with, e.g., [26].”

For the input quantization case, a Filippov solution P(t) exists and is absolutely continuous on [0,∞), and there exists at least one weak solution u(x,t) ∈ L∞([0,D]×R+;R) for the transport PDE part, yielding a solution X(t) ∈ AC(R+;R^n).

“Thus, a Filippov solution P ( t ) exists (under the local Lipschitzness assumptions on f and κ ) by, e.g., [14], which is absolutely continuous on [0 , ∞ ) .”