Regular Optimal Control Problems with Quadratic Final Penalties

VICENTE COSTANZA

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UMA

Lugar: Buenos Aires; Año: 2008 vol. 49 p. 43 - 56

0041-6932

Hamiltonfs canonical equations (HCEs) have played a central role in Mechanics after (i) their equivalence with the principle of least action, and (ii) the variational calculus leading to the Euler-Lagrange equation, were established and applied (see [1]). Also, since the foundational work of Pontryagin [22], HCEs have been at the core of modern optimal control theory. When the problem concerning an n-dimensional control system and an additive cost objective is regular [19], i.e. when the Hamiltonian H(t, x, , u) of the problem is smooth enough and can be uniquely optimized with respect to u at a control value u0(t, x, ) (depending on the remaining variables), then HCEs appear as a set of 2n ordinary differential equations whose solutions are optimal state-costate time trajectories. Concerning the infinite-horizon bilinear-quadratic regulator and change of setpoint servo problems, there exists a recent attempt to find the missing initial condition for the costate variable, based on a state-dependent (generalized) algebraic Riccati equation (GARE) with solution P‡(x), which allows to integrate the HCEs on-line with the underlying control process [9]. The same approach in a finite time-domain leads to a first-order partial differential equation (PDE) called eGen- eralized Differential Riccati Equationf (GDRE) (see [3], [6], [11]) for a time-state dependent matrix P(t, x), whose solution allows to calculate the missing initial costate (0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. lim-dimensional control system and an additive cost objective is regular [19], i.e. when the Hamiltonian H(t, x, , u) of the problem is smooth enough and can be uniquely optimized with respect to u at a control value u0(t, x, ) (depending on the remaining variables), then HCEs appear as a set of 2n ordinary differential equations whose solutions are optimal state-costate time trajectories. Concerning the infinite-horizon bilinear-quadratic regulator and change of setpoint servo problems, there exists a recent attempt to find the missing initial condition for the costate variable, based on a state-dependent (generalized) algebraic Riccati equation (GARE) with solution P‡(x), which allows to integrate the HCEs on-line with the underlying control process [9]. The same approach in a finite time-domain leads to a first-order partial differential equation (PDE) called eGen- eralized Differential Riccati Equationf (GDRE) (see [3], [6], [11]) for a time-state dependent matrix P(t, x), whose solution allows to calculate the missing initial costate (0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. limH(t, x, , u) of the problem is smooth enough and can be uniquely optimized with respect to u at a control value u0(t, x, ) (depending on the remaining variables), then HCEs appear as a set of 2n ordinary differential equations whose solutions are optimal state-costate time trajectories. Concerning the infinite-horizon bilinear-quadratic regulator and change of setpoint servo problems, there exists a recent attempt to find the missing initial condition for the costate variable, based on a state-dependent (generalized) algebraic Riccati equation (GARE) with solution P‡(x), which allows to integrate the HCEs on-line with the underlying control process [9]. The same approach in a finite time-domain leads to a first-order partial differential equation (PDE) called eGen- eralized Differential Riccati Equationf (GDRE) (see [3], [6], [11]) for a time-state dependent matrix P(t, x), whose solution allows to calculate the missing initial costate (0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. limu at a control value u0(t, x, ) (depending on the remaining variables), then HCEs appear as a set of 2n ordinary differential equations whose solutions are optimal state-costate time trajectories. Concerning the infinite-horizon bilinear-quadratic regulator and change of setpoint servo problems, there exists a recent attempt to find the missing initial condition for the costate variable, based on a state-dependent (generalized) algebraic Riccati equation (GARE) with solution P‡(x), which allows to integrate the HCEs on-line with the underlying control process [9]. The same approach in a finite time-domain leads to a first-order partial differential equation (PDE) called eGen- eralized Differential Riccati Equationf (GDRE) (see [3], [6], [11]) for a time-state dependent matrix P(t, x), whose solution allows to calculate the missing initial costate (0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. limn ordinary differential equations whose solutions are optimal state-costate time trajectories. Concerning the infinite-horizon bilinear-quadratic regulator and change of setpoint servo problems, there exists a recent attempt to find the missing initial condition for the costate variable, based on a state-dependent (generalized) algebraic Riccati equation (GARE) with solution P‡(x), which allows to integrate the HCEs on-line with the underlying control process [9]. The same approach in a finite time-domain leads to a first-order partial differential equation (PDE) called eGen- eralized Differential Riccati Equationf (GDRE) (see [3], [6], [11]) for a time-state dependent matrix P(t, x), whose solution allows to calculate the missing initial costate (0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. limP‡(x), which allows to integrate the HCEs on-line with the underlying control process [9]. The same approach in a finite time-domain leads to a first-order partial differential equation (PDE) called eGen- eralized Differential Riccati Equationf (GDRE) (see [3], [6], [11]) for a time-state dependent matrix P(t, x), whose solution allows to calculate the missing initial costate (0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. limeGen- eralized Differential Riccati Equationf (GDRE) (see [3], [6], [11]) for a time-state dependent matrix P(t, x), whose solution allows to calculate the missing initial costate (0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. lim(GDRE) (see [3], [6], [11]) for a time-state dependent matrix P(t, x), whose solution allows to calculate the missing initial costate (0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. limP(t, x), whose solution allows to calculate the missing initial costate (0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. lim(0) = 2P(0, x0)x0 and exhibits, for S = 0, a limiting behavior (see [19]) similar to that of linear systems with the same cost, i.e. lim T¨‡¨‡ P(0, x) = P‡(x) , (1) where T is the duration of each optimization process.(0, x) = P‡(x) , (1) where T is the duration of each optimization process.T is the duration of each optimization process.