Maximum principle for optimal boundary control of the Kuramoto–Sivashinsky equation

Necessary and Sufficient Conditions of Optimality for a Damped Hyperbolic Equation in One-Space Dimension

“…Under proper convexity assumption on state variable, it is proved thatthe maximum principle is also sufficient condition for the controls to be optimal . In [1][2][3][4][5][6][7], the systems under consideration include only one control function. Also, the necessary and sufficient conditions for optimality are given for a single hyperbolic differential equation without damping term subject to the homogeneous boundary conditions in onespace dimension.…”

“…Also, the necessary and sufficient conditions for optimality are given for a single hyperbolic differential equation without damping term subject to the homogeneous boundary conditions in onespace dimension. But, in the present paper, as an original contribution to literature, the necessary and sufficient optimality conditions obtained in [1][2][3][4][5][6][7] are generalized for a general class of damped hyperbolic equation involving damping and several control functions subject to nonhomogeneous boundary conditions in one space dimension.…”

“…For a general control problem formulated in terms of a differential inclusion, maximum principle is given by weak pseudo-Lipschitz behavior that is postulated on the underlying multifunction [5]. Necessary condition in the form of Pontryagin maximum principle by adopting the Dubovitskiy-Milyutin functional analytical approach is derived in [6]. In [7], Barnes presents a maximum principle as a necessary condition for optimal control of vibrating system that is modeled by second order linear hyperbolic PDE where completeness assumption is dropped and a regular point is used.…”

“…Other related theoretical studies about the maximum principle in the literature are available such as [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. In [1][2][3][4][5][6][7], the optimal necessary and sufficient conditions are derived for a single partial differential equation without damping term subject to the homogeneous boundary conditions in one space dimension.…”

Optimal boundary control of a continuum model for a highly re-entrant manufacturing system

“…Nowadays, it is believed that this method is especially suited for dealing with optimal control problems with multiple inequality constraints, whereas it is exactly one of the difficulties in optimal control theory. Many other optimal control approaches share this weakness in this regard (see, e.g., Chan and Guo, 1989; Dmitruk and Osmolovskii, 2014; Girsanov, 1972; Isac and Khan, 2008; Osmolovskii, 2018; Pedregal, 2017; Sun, 2010). Even for optimal control problems with time delay, there are also many successful examples that utilize the Dubovitskii–Milyutin method, such as that presented by Kotarski (1997).…”