Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems

Mikhail I. Gomoyunov

Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems

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Pith's one-line read Dynamic programming holds for fractional-order optimal control when the value functional is defined on trajectory histories, and a new calculus of coinvariant derivatives turns this principle into a Hamilton–Jacobi–Bellman equation.

desk verdict History-dependent DPP for Caputo fractional control is correct and unconditional; the HJB half is honest but applies only to a restricted, non-generic smooth class. read the letter →

arxiv 1908.01747 v1 pith:DBWEHRL5 submitted 2019-08-05 math.OC math.APmath.DS

Mikhail I. Gomoyunov This is my paper · ORCID

classification math.OCmath.APmath.DS MSC 26A3334A0849L2035F21

keywords optimalcontrolfractionalderivativesCaputoderivativedynamicprogrammingprincipleHamilton-Jacobi-Bellmanequationcoinvariantfeedbackhistory-dependentvalue

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows that dynamic programming works for fractional-order optimal control—but only if the 'position' of the system is taken to be the entire history of the motion up to the current time, not just the current state. The reason is that the Caputo derivative is nonlocal: its value at time $t$ depends on all earlier values, so any sub-problem starting at $t$ must be fed the past trajectory. With the value as a functional $\rho(t,w(\cdot))$ on histories, the paper proves the dynamic programming principle in full generality (Theorem 6.1), and then introduces a tailored calculus of 'coinvariant derivatives' of order $\alpha$ to write an associated Hamilton–Jacobi–Bellman equation. Under a ci-smoothness assumption, the value functional satisfies this equation, and any solution of the HJB Cauchy problem with the right terminal condition equals the value and yields an optimal feedback control strategy. The paper's concluding section notes the main limitation: the value may not actually be ci-smooth, so the HJB connection is conditional.

What carries the argument

The load-bearing object is the fractional coinvariant derivative. For a functional $\varphi(t,w(\cdot))$ on histories, $\varphi$ is ci-differentiable of order $\alpha$ at $(t,w(\cdot))$ if there exist $\partial^\alpha_t \varphi(t,w(\cdot)) \in \mathbb{R}$ and $\nabla^\alpha \varphi(t,w(\cdot)) \in \mathbb{R}^n$ such that for every extension $x(\cdot)$ of the history $w(\cdot)$ and every $\tau>t$, $\varphi(\tau,x_\tau(\cdot)) - \varphi(t,w(\cdot)) = \partial^\alpha_t \varphi\,(\tau-t) + \langle\nabla^\alpha \varphi,\, (I^{1-\alpha}(x(\cdot)-x(0)))(\tau) - (I^{1-\alpha}(w(\cdot)-w(0)))(t)\rangle + o(\tau-t)$. Interpreting the difference of fractional integrals as $\int_t^\tau (C D^\alpha x)(\xi)\,d\xi$, this yields the clean total-derivative formula $\frac{d}{d\tau}\varphi(\tau,x_\tau(\cdot)) = \partial^\alpha_t \varphi + \langle\nabla^\alpha \varphi,\, (C D^\alpha x)(\tau)\rangle$ along motions (Lemma 9.2). This formula is what converts the dynamic programming principle into the infinitesimal HJB equation, and it also makes the extremal-shift feedback construction possible.

What would settle it

Use the scalar system $(C D^\alpha x)(\tau)=\Gamma(\alpha+1)u(\tau)$, $|u(\tau)|\leq 1$, $\tau\in[0,T]$, with terminal cost $\sigma(x(T))=|x(T)|$ instead of $x^2$. The value becomes $\rho(t,w(\cdot)) = \max\{0,\, |\rho_*(t,w(\cdot))| - (T-t)^\alpha\}$, where $\rho_*$ is the ci-smooth functional computed in Section 12; at any history with $|\rho_*(t,w(\cdot))| = (T-t)^\alpha$ this functional has a kink and is not ci-differentiable of order $\alpha$. This directly violates the hypothesis of Theorems 10.1 and 11.1, showing that the dynamic programming principle alone does not produce the HJB equation.

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Extended reading notes

Core claim

For the Bolza problem with Caputo dynamics $(C D^\alpha x)(\tau)=f(\tau,x(\tau),u(\tau))$, the value functional $\rho(t,w(\cdot)) = \inf_{u\in U(t,T)} \big[\sigma(x(T))+\int_t^T \chi(\tau,x(\tau),u(\tau))\,d\tau\big]$ over histories $w(\cdot) \in AC^\alpha([0,t],\mathbb{R}^n)$ satisfies, for every intermediate time $\theta$, the dynamic programming principle $\rho(t,w(\cdot)) = \inf_{u\in U(t,\theta)} \big(\rho(\theta,x(\cdot)) + \int_t^\theta \chi(\tau,x(\tau),u(\tau))\,d\tau\big)$, where $x(\cdot)$ is the motion that continues the history $w(\cdot)$ under $u$. This principle is proved without extra assumptions. Using a new notion of coinvariant (ci-) differentiation of order $\alpha$, the paper associates the problem with the Hamilton–Jacobi–Bellman equation $\partial^\alpha_t \varphi(t,w(\cdot)) + H(t,w(t),\nabla^\alpha \varphi(t,w(\cdot))) = 0$, with the Hamiltonian $H(\tau,x,s)=\min_{u\in P}(\langle s,f(\tau,x,u)\rangle + \chi(\tau,x,u))$. If the value functional is ci-smooth of order $\alpha$, it solves this equation (Theorem 10.1); conversely, any ci-smooth solution with terminal condition $\varphi(T,w(\cdot))=\sigma(w(T))$ coincides with the value functional, and the strategy $U^\circ(t,w(\cdot)) \in \arg\min_{u\in P}\big(\langle\nabla^\alpha \varphi(t,w(\cdot)), f(t,w(t),u)\rangle + \chi(t,w(t),u)\big)$ is optimal (Theorem 11.1).

Load-bearing premise

The HJB and feedback theorems assume the value functional is ci-smooth of order $\alpha$, meaning it is ci-differentiable at every interior position with continuous ci-derivative and ci-gradient; the paper proves continuity of the value but not this differentiability, and Section 13 explicitly concedes that the value functional may fail to be ci-smooth.

Editorial extensions

If this is right

The dynamic programming principle (Theorem 6.1) is unconditional: for any admissible control split at an intermediate time $\theta$, the value from $t$ equals the infimum of the running cost on $[t,\theta]$ plus the value from the resulting history at $\theta$.
If the value functional is ci-smooth of order $\alpha$, it solves the Hamilton–Jacobi–Bellman equation (10.1), so in that case optimal control problems reduce to solving a single functional PDE.
Conversely, a ci-smooth solution of the HJB Cauchy problem with terminal condition $\sigma(w(T))$ is exactly the value functional, making the HJB equation equivalent to the optimal control problem in the smooth case.
Given such a solution $\varphi$, the feedback strategy $U^\circ(t,w(\cdot)) = \arg\min_{u\in P}\big(\langle\nabla^\alpha \varphi(t,w(\cdot)), f(t,w(t),u)\rangle + \chi(t,w(t),u)\big)$ is optimal in the positional sense: for any sufficiently fine partition, the stepwise control law $\{U^\circ,\Delta\}$ produces $\varepsilon$-optimal controls.
Corollary 11.4: when the value functional itself is ci-smooth, the strategy built from its ci-gradient by the same extremal shift is optimal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The gap the paper leaves open is exactly the differentiability of $\rho$; a natural next step is a viscosity or minimax theory for equation (10.1) on the history space $\mathcal{G}$, and the compactness results in Section 10 (Proposition 10.2) provide the infrastructure for such a theory.
Because the DPP is unconditional, it can be used as a correctness test for numerical schemes: any discretization that ignores the history dependence of the value will fail Bellman's recursion, so practical algorithms for fractional optimal control should feed the whole past trajectory into the value update.
The same ci-derivative calculus can be imported into stability analysis: Lemma 9.2 offers a total-derivative formula for Lyapunov–Krasovskii functionals on histories, which sidesteps the chain-rule difficulties that arise with functions $V(t,x(t))$ in fractional systems.
A testable extension: compute $\rho(t,w(\cdot))$ for the Section 12 example with terminal cost $|x(T)|$ instead of $x^2(T)$; the value has a kink at $|\rho_*| = (T-t)^\alpha$, localizing exactly where viscosity-type solutions would be needed and providing a benchmark problem for a nonsmooth HJB theory.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: Theorem 6.1, the dynamic programming principle for history-dependent values, is the real result and it is correct. The HJB part is conditional, and the smoothness assumption is not a minor technicality: it fails for an elementary admissible problem. The author admits this in Section 13, so it is a limitation, not a hidden error.

What is new: earlier fractional HJB work used state-only value functions, which are wrong for Caputo dynamics because of memory. The history-dependent value functional ρ(t,w(·)) fixes that, and the Section 4 example convincingly shows why a state-only position fails. The DPP proof is a standard argument, but the setup is careful: positions are histories, the semigroup property is used explicitly, and Proposition 7.1 supplies the uniform Hölder and Lipschitz bounds needed later. Continuity of ρ (Theorem 8.3) is a genuine piece of work. The fractional coinvariant derivative is a natural adaptation of ci-calculus to the fractional setting, and Lemma 9.2's total derivative formula is the useful payoff.

Where the soft spots are: The HJB verification theorems (10.1 and 11.1) require ci-smoothness of the value functional. This is not merely unverified. Take f ≡ 0, χ ≡ 0, and terminal cost σ(x) = |x|. Then ρ(t,w) = |ρ*(t,w)|, where ρ* is a linear functional of the history. At a history with ρ* = 0, the ci-expansion would force the map l ↦ |⟨s,l⟩| to be linear in l, which is impossible; even in one dimension the increment behaves like |l|(τ−t)^α, not linearly in τ−t. So the HJB equation does not apply to a perfectly admissible problem. The author explicitly acknowledges that the value may lack smoothness and defers to future viscosity-type theory, so the theorems are honestly stated; but anyone citing the HJB result should know it covers a restricted class. Lemma 9.2's Dini theorem step is terse and could use expansion. The citation pattern leans on the author's own prior result for existence, uniqueness, and semigroup properties, but that is a published adjacent result, not circular.

Who it is for: people working on fractional optimal control, and anyone connecting fractional dynamics to the functional-differential HJB tradition. It deserves a serious referee; I would recommend accept with minor revisions, mainly asking the authors to make the non-genericity of ci-smoothness visible earlier in the abstract and introduction.

Referee Report

0 major / 5 minor

Summary. The paper studies a Bolza-type optimal control problem for systems governed by a Caputo fractional differential equation of order α ∈ (0,1). Because the Caputo derivative is nonlocal, the author argues in Section 4 that the value of the problem must be viewed as a functional of the entire history w(·) on [0,t], not merely of the current state w(t). The main results are: (i) the dynamic programming principle (Theorem 6.1), proved directly for the history-dependent value functional; (ii) continuity of the value functional on the space of positions with a Hausdorff-type metric (Theorem 8.3); (iii) a new notion of fractional coinvariant (ci-) derivatives and a corresponding Hamilton-Jacobi-Bellman equation (Section 9 and Eq. (10.1)); (iv) a verification theorem stating that a ci-smooth value functional satisfies the HJB equation (Theorem 10.1) and that a ci-smooth solution of the HJB equation with the natural terminal condition coincides with the value and yields an optimal feedback strategy (Theorem 11.1). The paper closes with a fully worked example that is solved using the HJB framework.

Significance. If correct, the paper gives a clean and mathematically rigorous extension of dynamic programming and HJB methods to fractional-order control systems, and it is transparent about the history-dependent nature of the problem, which is an essential and non-obvious point. The direct proof of the dynamic programming principle is unconditional and does not rely on smoothness assumptions. The introduction of fractional ci-derivatives is natural and, as shown in Lemma 9.2, yields a simple formula for the total derivative of functionals along fractional trajectories, which is a useful tool for future work. The HJB verification results are explicitly conditional on ci-smoothness, and Section 13 correctly states that the value functional may fail to satisfy this assumption in general. This is a scope limitation rather than an internal inconsistency: the paper does not claim the HJB equivalence for the entire non-smooth problem class. The worked example in Section 12 is internally consistent and illustrates the theory well.

minor comments (5)

[Theorem 6.1, Eq. (6.2)] The notation ρ(θ,x(·)) is slightly ambiguous because x(·) denotes the motion on [0,θ] in the theorem statement, while elsewhere x(·) denotes a full trajectory on [0,T]; please clarify explicitly that ρ(θ,x(·)) means ρ(θ,x_θ(·)), the value functional evaluated at the history of x on [0,θ].
[Lemma 9.2, proof] The application of Dini's theorem is very terse: the proof bounds the upper right derivative of ω, but to conclude two-sided Lipschitz continuity one should note that the same argument applied to −ω gives the lower bound; adding one sentence would make the argument fully transparent.
[Section 13] The caveat that the value functional may not be ci-smooth is stated only in the concluding section; because it is essential for interpreting Theorems 10.1 and 11.1, this limitation should also be signaled in the introduction or in the abstract's discussion of the HJB results.
[Proof of Theorem 10.1] The phrase "Formally extending the motion x(·) up to T" is vague; please specify that the extension is obtained by continuing with the same constant control on [t+δ,T], after which Lemma 9.2 is applied on [t,t+δ].
[Section 12, formulas for ci-derivatives] In the displayed formulas for ∂_t^α φ and ∇^α φ in the first case, the absence of parentheses around "2ρ*(t,w(·)) + (T−t)^α" makes the formulas ambiguous; adding parentheses will improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the DPP is proved from the definition of the value functional, and the HJB theorems are conditional verification results; the acknowledged smoothness limitation is a completeness caveat, not a circular step.

full rationale

Neither the dynamic programming principle nor the Hamilton-Jacobi-Bellman results reduce to their own inputs by construction. The value functional is defined independently as an infimum over controls in (5.5), and Theorem 6.1 proves the DPP from that definition together with the semigroup property of motions, without assuming the equality (6.2). The ci-derivative notion in Section 9 is a new technical definition, and Theorem 10.1 derives the HJB equation (10.1) from the DPP and the chain-rule Lemma 9.2, while Theorem 11.1 proves the converse verification statement for any ci-smooth solution. No parameter is fitted and no quantity called a prediction is extracted from the same data that defines it. The paper does rely on the author's own prior results, notably [15, Proposition 2] for existence, uniqueness, and semigroup properties of motions and [13, Assertion 7] for compactness, but these are ancillary technical facts that do not contain the DPP or HJB theorems; they are independent support for standard building blocks rather than load-bearing circular citations. The most substantive caveat is that the HJB equivalence is conditional on ci-smoothness of the value functional, and Section 13 explicitly states: the value functional may not possess the specified smoothness properties. This is a genuine limitation affecting the completeness of the HJB connection for the full problem class, but it is not circularity: the conditional theorems remain valid under their stated assumption, and no derivation is secretly identical to an input by definition. Accordingly, the appropriate finding is no significant circularity, with only minor non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the only numerical constants come from standard fractional calculus. The paper introduces a new differential calculus for history functionals, but that is a definition rather than a postulated physical entity. The main structural load is the ci-smoothness hypothesis on the value functional and a prior existence/semigroup result cited from the author's own work.

assumptions (4)

domain assumption Assumption 3.1 (f continuous, locally Lipschitz in x, sublinear growth) and Assumption 3.2 (σ and χ continuous), with a compact control set P.
Standard well-posedness and boundedness hypotheses for the optimal control problem and the value functional.
standard math Standard fractional calculus facts: Riemann-Liouville integral Holder estimate (2.1), Caputo representation (2.5)-(2.6), and the Mittag-Leffler Gronwall lemma (Lemma 7.3).
Used in Sections 7 and 8 for a priori bounds, Holder continuity of motions, and continuity estimates of the value functional.
domain assumption Cited result [15, Proposition 2]: for any initial history (t,w(·)) and admissible control, there is a unique motion of system (3.1) satisfying the semigroup and concatenation property.
Defines sub-problem motions and is needed for the dynamic programming principle and for the value functional (5.5); cited rather than proved in this paper.
ad hoc to paper ci-smoothness of the value functional ρ, or of a candidate solution φ, as defined in Section 9 and assumed in Theorems 10.1 and 11.1.
This is the core regularity condition for the HJB results. The paper proves continuity but not this differentiability, and Section 13 states it may fail in general.

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Pith. "Pith review of Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems." pith.science (2026). https://pith.science/paper/DBWEHRL5

@misc{pith2026190801747,
  author       = {Pith},
  title        = {Pith review of: Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DBWEHRL5}},
  note         = {Machine review of arXiv:1908.01747}
}

abstract

We consider a Bolza-type optimal control problem for a dynamical system described by a fractional differential equation with the Caputo derivative of an order $\alpha \in (0, 1)$. The value of this problem is introduced as a functional in a suitable space of histories of motions. We prove that this functional satisfies the dynamic programming principle. Based on a new notion of coinvariant derivatives of the order $\alpha$, we associate the considered optimal control problem with a Hamilton-Jacobi-Bellman equation. Under certain smoothness assumptions, we establish a connection between the value functional and a solution to this equation. Moreover, we propose a way of constructing optimal feedback controls. The paper concludes with an example.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 35 canonical work pages

[1]

Agarw al, D

R. Agarw al, D. O’Regan, and S. Hristova , Stability of Caputo fractional diﬀerential equa- tions by Lyapunov functions , Appl. Math., 60 (2015), pp. 653–676, https://doi.org/10. 1007/s10492-015-0116-4

work page 2015
[2]

Aguila-Camacho, M

N. Aguila-Camacho, M. A. Duarte-Mermoud, and J. A. Gallegos , Lyapunov functions for fractional order systems , Comm. Nonlin. Sci. Num. Simul., 19 (2014), pp. 2951–2957, https://doi.org/10.1016/j.cnsns.2014.01.022

work page doi:10.1016/j.cnsns.2014.01.022 2014
[3]

A. A. Alikhanov , A priori estimates for solutions of boundary value problems for fractional–order equations , Diﬀ. Eqs., 46 (2010), pp. 660–666, https://doi.org/10.1134/ s0012266110050058. DPP AND HJB EQUATIONS FOR FRACTIONAL-ORDER SYSTEMS 25

work page 2010
[4]

Aubin and G

J.-P. Aubin and G. Haddad , History path dependent optimal control and portfolio val- uation and management , Positivity, 6 (2002), pp. 331–358, https://doi.org/10.1023/A: 1020244921138

doi:10.1023/a: 2002
[5]

Bayraktar and C

E. Bayraktar and C. Keller , Path-dependent Hamilton–Jacobi equations in inﬁnite dime n- sions, J. Funct. Anal., 275 (2018), pp. 2096–2161, https://doi.org/10.1016/j.jfa.2018.07. 010

work page doi:10.1016/j.jfa.2018.07 2018
[6]

Bellman , Dynamic programming, Princeton University Press, 1957

R. Bellman , Dynamic programming, Princeton University Press, 1957

work page 1957
[7]

A. M. Bruckner , Diﬀerentiation of real functions , vol. 659 of Lecture Notes in Mathematics, Springer, ﬁrst ed., 1978, https://doi.org/10.1007/bfb0069821

work page doi:10.1007/bfb0069821 1978
[8]

Burton , Fractional diﬀerential equations and Lyapunov functional s, Nonlinear Anal., 74 (2011), pp

T. Burton , Fractional diﬀerential equations and Lyapunov functional s, Nonlinear Anal., 74 (2011), pp. 5648–5662, https://doi.org/10.1016/j.na.2011.05.050

work page doi:10.1016/j.na.2011.05.050 2011

Show all 44 references

[9]

N. D. Cong and H. T. Tuan , Generation of nonlocal fractional dynamical systems by fra ctional diﬀerential equations , J. Integral Equations Applications, 29 (2017), pp. 585–60 8, https:// doi.org/10.1216/JIE-2017-29-4-585

2017 doi
[10]

Diethelm , The analysis of fractional diﬀerential equations: an appli cation-oriented exposi- tion using diﬀerential operators of Caputo type , vol

K. Diethelm , The analysis of fractional diﬀerential equations: an appli cation-oriented exposi- tion using diﬀerential operators of Caputo type , vol. 2004 of Lecture Notes in Mathematics, Springer, 2010, https://doi.org/10.1007/978-3-642-14574-2

2004 doi
[11]

Dupire, Functional Itˆ o calculus, (2009), https://ssrn.com/abstract=1435551 (accessed 2019- 07-22)

B. Dupire, Functional Itˆ o calculus, (2009), https://ssrn.com/abstract=1435551 (accessed 2019- 07-22)

2009
[12]

Flores-Tlacuahuac and L

A. Flores-Tlacuahuac and L. T. Biegler , Optimization of fractional order dynamic chem- ical processing systems , Ind. Eng. Chem. Res., 53 (2014), pp. 5110–5127, https://doi.org/ 10.1021/ie401317r

2014 doi
[13]

M. I. Gomoyunov , Extremal shift to accompanying points in a positional diﬀer ential game for a fractional-order system , Trudy Inst. Mat. i Mekh. UrO RAN, 25 (2019), pp. 11–34, https://doi.org/10.21538/0134-4889-2019-25-1-11-34 . (in Russian)

2019 doi
[14]

M. I. Gomoyunov , Fractional derivatives of convex Lyapunov functions and co ntrol problems in fractional order systems , Fract. Calc. Appl. Anal., 21 (2019), pp. 1238–1261, https:// doi.org/10.1515/fca-2018-0066

2019 doi
[15]

M. I. Gomoyunov , Solution to a zero-sum diﬀerential game with fractional dyn amics via approximations, Dyn. Games Appl., (2019), https://doi.org/10.1007/s13235-019-00320-4

2019 doi
[16]

M. I. Gomoyunov and A. R. Plaksin , On Hamilton–Jacobi equations for neutral-type dif- ferential games , IF AC-PapersOnLine, 51 (2018), pp. 171–176, https://doi.org/10.1016/j. ifacol.2018.07.218

2018 doi
[17]

Hajipour, M

A. Hajipour, M. Hajipour, and D. Baleanu , On the adaptive sliding mode controller for a hyperchaotic fractional-order ﬁnancial system , Physica A, 497 (2018), pp. 139–153, https://doi.org/10.1016/j.physa.2018.01.019

2018 doi
[18]

Jumarie , Fractional Hamilton–Jacobi equation for the optimal contr ol of nonrandom frac- tional dynamics with fractional cost function , J

G. Jumarie , Fractional Hamilton–Jacobi equation for the optimal contr ol of nonrandom frac- tional dynamics with fractional cost function , J. Appl. Math. Comput., 23 (2007), pp. 215– 228, https://doi.org/10.1007/BF02831970

2007 doi
[19]

Kaczorek , Minimum energy control of fractional positive electrical c ircuits with bounded inputs, Circuits Syst

T. Kaczorek , Minimum energy control of fractional positive electrical c ircuits with bounded inputs, Circuits Syst. Signal Process., 35 (2016), pp. 1815–1829, https://doi.org/10.1007/ s00034-015-0181-7

2016
[20]

H. Kaise , Path-dependent diﬀerential games of inf-sup type and Isaac s partial diﬀerential equations, in Proceedings of the 54-th IEEE Conference on Decision and Control (CDC), Osaka, Japan, Dec 2015, pp. 1972–1977, https://doi.org/10.1109/CDC.2015.7402496

2015
[21]

Kaise, T

H. Kaise, T. Kato, and Y. Takahashi , Hamilton–Jacobi partial diﬀerential equations with path-dependent terminal costs under superlinear Lagrangi ans, in Proceedings of the 23-rd International Symposium on Mathematical Theory of Network s and Systems (MTNS2018), Hong Kong Univ...

2018
[22]

Kheiri and M

H. Kheiri and M. Jafari , Optimal control of a fractional-order model for the HIV/AID S epidemic, Int. J. Biomath., 11 (2018), 1850086 (23 pages), https://doi.org/10.1142/ S1793524518500869

2018
[23]

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo , Theory and applications of fractional diﬀerential equations , vol. 204 of North-Holland Mathematics Studies, Elsevier, 2006

2006
[24]

A. V. Kim , Lyapunov 2-nd method for systems with aftereﬀect , Diﬀ. Eqs., 21 (1985), pp. 244– 249

1985
[25]

A. V. Kim , Functional diﬀerential equations: application of i-smoot h calculus, vol. 479 of Math- ematics and Its Applications, Springer, 1999, https://doi.org/10.1007/978-94-017-1630-7

1999 doi
[26]

A. N. Krasovskii and N. N. Krasovskii , Control under lack of information , Systems & Control: Foundations & Applications, Birkh¨ auser, 1995, https://doi.org/10.1007/ 978-1-4612-2568-3

1995
[27]

N. N. Krasovskii , Stability of motion: applications of Lyapunov’s second met hod to diﬀerential 26 M. I. GOMOYUNOV systems and equations with delay , Stanford University Press, 1963

1963
[28]

N. N. Krasovskii and A. I. Subbotin , Game-theoretical control problems, Springer Series in Soviet Mathematics, Springer, 1988

1988
[29]

N. Y. Lukoyanov , A Hamilton–Jacobi type equation in control problems with he reditary information, J. Appl. Math. Mech., 64 (2000), pp. 243–253, https://doi.org/10.1016/ S0021-8928(00)00046-0

2000
[30]

N. Y. Lukoyanov , Functional Hamilton–Jacobi type equations with ci-deriva tives in control problems with hereditary information , Nonlinear Funct. Anal. Appl., 8 (2003), pp. 535–555

2003
[31]

N. Y. Lukoyanov, Functional Hamilton–Jacobi equations and control problem s with hereditary information, Ural Federal University Publishing, 2011. (in Russian)

2011
[32]

N. Y. Lukoyanov, M. I. Gomoyunov, and A. R. Plaksin , Hamilton–Jacobi functional equa- tions and diﬀerential games for neutral-type systems , Dokl. Math., 96 (2017), pp. 654–657, https://doi.org/10.1134/S1064562417060114

2017 doi
[33]

Y. S. Osipov , On the theory of diﬀerential games of systems with aftereﬀec t, J. Appl. Math. Mech., 35 (1971), pp. 262–272, https://doi.org/10.1016/0021-8928(71)90032-3

1971 doi
[34]

Pham and J

T. Pham and J. Zhang , Two person zero-sum game in weak formulation and path depend ent Bellman–Isaacs equation , SIAM J. Control Optim., 52 (2014), pp. 2090–2121, https://doi. org/10.1137/120894907

2014 doi
[35]

A. R. Plaksin , On Hamilton–Jacobi–Bellman–Isaacs equation for time-del ay systems , IF AC- PapersOnLine, (2019). (accepted for publication)

2019
[36]

S. A. Rakhshan, S. Effati, and A. V. Kamyad , Solving a class of fractional optimal control problems by the Hamilton–Jacobi–Bellman equation , J. Vib. Control, 24 (2018), pp. 1741– 1756, https://doi.org/10.1177/1077546316668467

2018 doi
[37]

Razminia, M

A. Razminia, M. Asadizadehshiraz, and D. F. M. Torres , Fractional order version of the Hamilton–Jacobi–Bellman equation , J. Comput. Nonlinear Dynam., 14 (2018), 011005 (6 pages), https://doi.org/10.1115/1.4041912

2018 doi
[38]

S. G. Samko, A. A. Kilbas, and O. I. Marichev , Fractional integrals and derivatives: theory and applications , Gordon and Breach Science Publishers, 1993

1993
[39]

Saporito , Stochastic control and diﬀerential games with path-depend ent inﬂuence of con- trols on dynamics and running cost , SIAM J

Y. Saporito , Stochastic control and diﬀerential games with path-depend ent inﬂuence of con- trols on dynamics and running cost , SIAM J. Control Optim., 57 (2019), pp. 1312–1327, https://doi.org/10.1137/18M1186186

2019 doi
[40]

V. E. Tarasov , On chain rule for fractional derivatives , Comm. Nonlin. Sci. Num. Simul., 30 (2016), pp. 1–4, https://doi.org/10.1016/j.cnsns.2015.06.007

2016 doi
[41]

V. E. Tarasov , No nonlocality. No fractional derivative , Comm. Nonlin. Sci. Num. Simul., 62 (2018), pp. 157–163, https://doi.org/10.1016/j.cnsns.2018.02.019

2018 doi
[42]

Toledo-Hernandez, V

R. Toledo-Hernandez, V. Rico-Ramirez, R. Rico-Martinez, S. He rnandez-Castro, and U. M. Diwekar , A fractional calculus approach to the dynamic optimization of biological reactive systems. Part II: numerical solution of fractiona l optimal control problems , Chem. Eng. Sci., 1...

2014 doi
[43]

H. T. Tuan and H. Trinh , Stability of fractional–order nonlinear systems by Lyapun ov direct method, IET Control Theory & Applications, 12 (2018), pp. 2417–242 2, https://doi.org/ 10.1049/iet-cta.2018.5233

2018
[44]

Yong , Diﬀerential games: a concise introduction , W orld scientiﬁc, 2015, https://doi.org/ 10.1142/9121

J. Yong , Diﬀerential games: a concise introduction , W orld scientiﬁc, 2015, https://doi.org/ 10.1142/9121

2015 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Agarw al, D

R. Agarw al, D. O’Regan, and S. Hristova , Stability of Caputo fractional diﬀerential equa- tions by Lyapunov functions , Appl. Math., 60 (2015), pp. 653–676, https://doi.org/10. 1007/s10492-015-0116-4

work page 2015

[2] [2]

Aguila-Camacho, M

N. Aguila-Camacho, M. A. Duarte-Mermoud, and J. A. Gallegos , Lyapunov functions for fractional order systems , Comm. Nonlin. Sci. Num. Simul., 19 (2014), pp. 2951–2957, https://doi.org/10.1016/j.cnsns.2014.01.022

work page doi:10.1016/j.cnsns.2014.01.022 2014

[3] [3]

A. A. Alikhanov , A priori estimates for solutions of boundary value problems for fractional–order equations , Diﬀ. Eqs., 46 (2010), pp. 660–666, https://doi.org/10.1134/ s0012266110050058. DPP AND HJB EQUATIONS FOR FRACTIONAL-ORDER SYSTEMS 25

work page 2010

[4] [4]

Aubin and G

J.-P. Aubin and G. Haddad , History path dependent optimal control and portfolio val- uation and management , Positivity, 6 (2002), pp. 331–358, https://doi.org/10.1023/A: 1020244921138

doi:10.1023/a: 2002

[5] [5]

Bayraktar and C

E. Bayraktar and C. Keller , Path-dependent Hamilton–Jacobi equations in inﬁnite dime n- sions, J. Funct. Anal., 275 (2018), pp. 2096–2161, https://doi.org/10.1016/j.jfa.2018.07. 010

work page doi:10.1016/j.jfa.2018.07 2018

[6] [6]

Bellman , Dynamic programming, Princeton University Press, 1957

R. Bellman , Dynamic programming, Princeton University Press, 1957

work page 1957

[7] [7]

A. M. Bruckner , Diﬀerentiation of real functions , vol. 659 of Lecture Notes in Mathematics, Springer, ﬁrst ed., 1978, https://doi.org/10.1007/bfb0069821

work page doi:10.1007/bfb0069821 1978

[8] [8]

Burton , Fractional diﬀerential equations and Lyapunov functional s, Nonlinear Anal., 74 (2011), pp

T. Burton , Fractional diﬀerential equations and Lyapunov functional s, Nonlinear Anal., 74 (2011), pp. 5648–5662, https://doi.org/10.1016/j.na.2011.05.050

work page doi:10.1016/j.na.2011.05.050 2011