Optimal piecewise constant control of continuous time systems with time-varying delay

Banaś, Józef; Vacroux, A.G.

doi:10.1016/0005-1098(70)90029-4

Cited by 14 publications

(13 citation statements)

References 4 publications

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“…In this example we have A = t , A 1 =1, B = 1, Q f =0 and Q = R = 2. Thus the Pontryagin's maximum principle for the TDOCP () and () provides the following necessary conditions of optimality

({, \begin{matrix} arrayleft \\ \dot{x} (t) & = tx (t) + x (t - τ (t)) \\ - 0.5 λ (t) + v (t), 0 \leq t \leq 2, \\ \dot{λ} (t) & = - 2 x (t) - tλ (t) \\ - χ_{[0, 2 - τ (2)]} (t) \frac{λ (t + ρ (t))}{1 - \dot{τ} (t + ρ (t))}, 0 \leq t \leq 2, \\ x (t) & = 1, - τ (0) \leq t \leq 0, \\ λ (2) & = 0, \end{matrix})

where χ [0,2 − τ (2)] ( t ) is the characteristic function on [0,2 − τ (2)] and the lead function ρ (·) is determined as

ρ (t) = \sqrt{t^{2} + 2 t + 2}

. For h = 0.4,0.04,0.004, simulation results including the cost functional value and the elapsed CPU time are listed in Table .…”

Section: The Second‐order Two‐step Fd Methodsmentioning

confidence: 99%

“…In this section we shall be concerned with the analysis of the two-step FD method given by (9) for the system of ordinary differential equations (ODEs) (6). From Ref.…”

Section: Convergence Analysismentioning

confidence: 99%

“…Also, A ( t ), A 1 ( t ) and B ( t ) are continuous matrices of appropriate dimensions,

Q_{f} \in {double-struckR}^{n \times n}

and

Q \in {double-struckR}^{n \times n}

are positive semi‐definite matrices and

R \in {double-struckR}^{m \times m}

is a positive definite matrix. For the time‐delay optimal control problem (TDOCP) ()‐(), the Pontryagin's maximum principle provides the following necessary conditions of optimality

({, \begin{matrix} \dot{x} (t) & = A (t) x (t) + A_{1} (t) x (t - τ (t)) \\ - B (t) R^{- 1} B^{T} (t) λ (t), t_{0} \leq t \leq t_{f}, \\ \dot{λ} (t) & = - Qx (t) - A^{T} (t) λ (t) \\ - C (t) λ (t + ρ (t)), t_{0} \leq t \leq t_{f}, \end{matrix})

and

({, \begin{matrix} x (t) & = ϕ (t), t_{0} - τ (t_{0}) \leq t \leq t_{0}, \\ λ (t_{f}) & = Q_{f} x (t_{f}), \end{matrix})

where

λ (t) \in {double-struckR}^{n}

is the co‐state vector,

C (

…”

Section: Introductionmentioning

confidence: 99%

“…is introduced to take into account the functional dependence of the delay τ (.) on time; if s = t − τ ( t ), t 0 ≤ t ≤ t f , is solved for t , then ρ ( s ) is given by t = s + ρ ( s ) . Also, the optimal control law is

u^{*} (t) = - R^{- 1} B^{T} (t) λ (t), t_{0} \leq t \leq t_{f} .

The problem given by Eqs.…”

Section: Introductionmentioning

confidence: 99%

“…Also, A(t), A 1 (t) and B(t) are continuous matrices of appropriate dimensions, Q f ∈ R n×n and Q ∈ R n×n are positive semi-definite matrices and R ∈ R m×m is a positive definite matrix. For the time-delay optimal control problem (TDOCP) (1)- (2), the Pontryagin's maximum principle [6] provides the following necessary conditions of optimality…”

Section: Introductionmentioning

confidence: 99%

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An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

Jajarmi

Hajipour

2016

Asian Journal of Control

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In this paper, an efficient finite difference method is presented for the solution of time‐delay optimal control problems with time‐varying delay in the state. By using the Pontryagin's maximum principle, the original time‐delay optimal control problem is first transformed into a system of coupled two‐point boundary value problems involving both delay and advance terms. Then the derived system is converted into a system of linear algebraic equations by using a second‐order finite difference formula and a Hermite interpolation polynomial for the first‐order derivatives and delay terms, respectively. The convergence analysis of the proposed approach is provided. The new scheme is also successful for the optimal control of time‐delay systems affected by external persistent disturbances. Numerical examples are included to demonstrate the validity and applicability of the new technique. Some comparative results are included to illustrate the effectiveness of the proposed method.

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“…In this example we have A = t , A 1 =1, B = 1, Q f =0 and Q = R = 2. Thus the Pontryagin's maximum principle for the TDOCP () and () provides the following necessary conditions of optimality

({, \begin{matrix} arrayleft \\ \dot{x} (t) & = tx (t) + x (t - τ (t)) \\ - 0.5 λ (t) + v (t), 0 \leq t \leq 2, \\ \dot{λ} (t) & = - 2 x (t) - tλ (t) \\ - χ_{[0, 2 - τ (2)]} (t) \frac{λ (t + ρ (t))}{1 - \dot{τ} (t + ρ (t))}, 0 \leq t \leq 2, \\ x (t) & = 1, - τ (0) \leq t \leq 0, \\ λ (2) & = 0, \end{matrix})

where χ [0,2 − τ (2)] ( t ) is the characteristic function on [0,2 − τ (2)] and the lead function ρ (·) is determined as

ρ (t) = \sqrt{t^{2} + 2 t + 2}

. For h = 0.4,0.04,0.004, simulation results including the cost functional value and the elapsed CPU time are listed in Table .…”

Section: The Second‐order Two‐step Fd Methodsmentioning

confidence: 99%

“…In this section we shall be concerned with the analysis of the two-step FD method given by (9) for the system of ordinary differential equations (ODEs) (6). From Ref.…”

Section: Convergence Analysismentioning

confidence: 99%

“…Also, A ( t ), A 1 ( t ) and B ( t ) are continuous matrices of appropriate dimensions,

Q_{f} \in {double-struckR}^{n \times n}

and

Q \in {double-struckR}^{n \times n}

are positive semi‐definite matrices and

R \in {double-struckR}^{m \times m}

is a positive definite matrix. For the time‐delay optimal control problem (TDOCP) ()‐(), the Pontryagin's maximum principle provides the following necessary conditions of optimality

({, \begin{matrix} \dot{x} (t) & = A (t) x (t) + A_{1} (t) x (t - τ (t)) \\ - B (t) R^{- 1} B^{T} (t) λ (t), t_{0} \leq t \leq t_{f}, \\ \dot{λ} (t) & = - Qx (t) - A^{T} (t) λ (t) \\ - C (t) λ (t + ρ (t)), t_{0} \leq t \leq t_{f}, \end{matrix})

and

({, \begin{matrix} x (t) & = ϕ (t), t_{0} - τ (t_{0}) \leq t \leq t_{0}, \\ λ (t_{f}) & = Q_{f} x (t_{f}), \end{matrix})

where

λ (t) \in {double-struckR}^{n}

is the co‐state vector,

C (

…”

Section: Introductionmentioning

confidence: 99%

u^{*} (t) = - R^{- 1} B^{T} (t) λ (t), t_{0} \leq t \leq t_{f} .

The problem given by Eqs.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

Jajarmi

Hajipour

2016

Asian Journal of Control

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Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

2022

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In the mid‐1950s, Pontryagin et al. published a principle that became a fundamental concept in optimal control (OC) theory. The principle provides theoretical and practical methods to find the solution of OC problems, in particular, open‐loop control problems. In chemical engineering, the principle has played an important role as a decision making framework for more than 60 years. This study gathers the main contributions on the application of the Pontryagin's principle to the dynamic optimization of chemical processes. A concise overview of the optimality conditions for a wide class of constrained OC problems is provided. Numerical methods to solve the necessary conditions and strategies to address inequality constraints are summarized. The information and illustrative case study presented in this work can be used as a guide to implement the principle in different settings. Opportunities for further application of the principle in relevant chemical engineering problems are also discussed.

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Finite-time stability and optimal impulsive control for age-structured HIV model with time-varying delay and Lévy noise

Guo

Zhang

et al. 2021

Nonlinear Dyn

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Optimal piecewise constant control of continuous time systems with time-varying delay

Cited by 14 publications

References 4 publications

An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

Finite-time stability and optimal impulsive control for age-structured HIV model with time-varying delay and Lévy noise

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