Zermelo:
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permanent case : In this example, we investigate the Zermelo navigation problem under the assumption of permanent control. More precisely, the control input
$u(\cdot)$ can be adjusted at any time$t \in [0,8]$ . -
contro loss case 1: one region of control loss : In this example, we consider the Zermelo navigation problem with a state space divided into two regions: one of type C (control region) and one of type NC (loss of control region). Specifically, the control input
$u(\cdot)$ can only be modified when$x(\cdot)$ belongs to the region of type C. However, when$x(\cdot)$ belongs to the region of type NC, the control is constrained to a constant value to be determined. -
control loss case 1: via augmentaion : In this example, we investigate the same problem as in contro loss case 1: one region of control loss only we use the technique of augmentation.
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contro loss case 2: two regions of control loss : In this example, we consider the Zermelo navigation problem with a state space divided into three regions: one of type C (control region) and two of type NC (loss of control region). Specifically, the control input
$u(\cdot)$ can only be modified when$x(\cdot)$ belongs to the region of type C. However, when$x(\cdot)$ belongs to the regions of type NC, the control is constrained to a constant value to be determined (this constant value may differ between the two regions.)
Harmonic oscillator:
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permanent case : In this example, we investigate the harmonic oscillator problem under the assumption of permanent control. More precisely, the control input
$u(\cdot)$ can be adjusted at any time$t$ . -
control loss case: minimum time problem : In this example, we consider the harmonic oscillator problem, where we minimize the final time to reach the origin, with a state space divided into two regions: one of type C (control region) and one of type NC (loss of control region). Specifically, the control input
$u(\cdot)$ can only be modified when$x(\cdot)$ belongs to the region of type C. However, when$x(\cdot)$ belongs to the region of type NC, the control is constrained to a constant value to be determined. -
contro loss case 2: minimum energy problem : In this example, we consider the harmonic oscillator problem, where we minimize the energy to reach the origin, with a state space divided into two regions: one of type C (control region) and one of type NC (loss of control region). Specifically, the control input
$u(\cdot)$ can only be modified when$x(\cdot)$ belongs to the region of type C. However, when$x(\cdot)$ belongs to the region of type NC, the control is constrained to a constant value to be determined.
Double integrator:
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permanent case : In this example, we investigate the double integrator problem under the assumption of permanent control. More precisely, the control input
$u(\cdot)$ can be adjusted at any time$t$ . -
control loss case : In this example, we consider the double integrator problem with a state space divided into two regions: one of type C (control region) and one of type NC (loss of control region). Specifically, the control input
$u(\cdot)$ can only be modified when$x(\cdot)$ belongs to the region of type C. However, when$x(\cdot)$ belongs to the region of type NC, the control is constrained to a constant value to be determined.