INTEC   05402
INSTITUTO DE DESARROLLO TECNOLOGICO PARA LA INDUSTRIA QUIMICA
Unidad Ejecutora - UE
artículos
Título:
Regular Optimal Control Problems with Quadratic Final Penalties
Autor/es:
VICENTE COSTANZA
Revista:
REVISTA DE LA UNIóN MATEMáTICA ARGENTINA
Editorial:
UMA
Referencias:
Lugar: Buenos Aires; Año: 2008 vol. 49 p. 43 - 56
ISSN:
0041-6932
Resumen:
Hamiltonfs 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 Equationf (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 Equationf (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 Equationf (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 Equationf (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 Equationf (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 Equationf (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.
limeGen-
eralized Differential Riccati Equationf (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.