Control results with overdetermination condition for higher order dispersive system

Control of Kawahara equation with overdetermination condition: The unbounded cases

“… The integral overdetermination condition is effective and gives good control properties. This kind of condition was first applied in the inverse problem (see, e.g., [13]) and, more recently, in control theory [12, 14, 15]. One should be able of controlling the system, when the control acts in $$$ \left[0,T\right] $$$, on an unbounded domain, which is new for the Kawahara equation. We are also able to prove the existence of a minimal $$$ T>0 $$$ such that the overdetermination condition is still verified; however, we believe that this time is not optimal. …”

“…Our main focus is to investigate a type of controllability for the higher-order KdV type equation. We will continue working with an integral overdetermination condition started in [12] however in another framework, to be precise, on an unbounded domain. To do that, consider the initial boundary value problem (IBVP)…”

“…• The integral overdetermination condition is effective and gives good control properties. This kind of condition was first applied in the inverse problem (see, e.g., [13]) and, more recently, in control theory [12,14,15]. • One should be able of controlling the system, when the control acts in [0, T], on an unbounded domain, which is new for the Kawahara equation.…”

“…$$ Our main focus is to investigate a type of controllability for the higher‐order KdV type equation. We will continue working with an integral overdetermination condition started in [12] however in another framework, to be precise, on an unbounded domain . To do that, consider the initial boundary value problem (IBVP) $$$$ \left\{\begin{array}{ll}{u}_t+\alpha {u}_x+\beta {u}_{xxx}+\xi {u}_{xxx xx}+u{u}_x={f}_0(t)g\left(t,x\right)& \mathrm{in}\kern0.5em \left[0,T\right]\times {\mathrm{\mathbb{R}}}^{+},\\ {}u\left(t,0\right)={h}_1(t),{u}_x\left(t,0\right)={h}_2(t)& \mathrm{on}\kern0.5em \left[0,T\right],\\ {}u\left(0,x\right)={u}_0(x)& \mathrm{in}\kern0.5em {\mathrm{\mathbb{R}}}^{+},\end{array}\right.$$…”

“…Finally, concerning a new tool to find control properties for dispersive systems, we can cite a recent work of the first two authors [12]. In this work, the authors showed a new type of controllability for a dispersive fifth‐order equation that models water waves, what they called overdetermination control problem .…”