Projected Evolutionary Lifting and Well-Posedness of Stationary Hamilton-Jacobi-Bellman Equations in Infinite Dimensions

Fabian Fuchs; Gabriele Bolli

arxiv: 2605.05028 · v2 · submitted 2026-05-06 · 🧮 math.OC · math.AP

Projected Evolutionary Lifting and Well-Posedness of Stationary Hamilton-Jacobi-Bellman Equations in Infinite Dimensions

Gabriele Bolli , Fabian Fuchs This is my paper

Pith reviewed 2026-05-08 17:17 UTC · model grok-4.3

classification 🧮 math.OC math.AP

keywords Hamilton-Jacobi-Bellman equationsinfinite-dimensional stochastic controlmild solutionsmaximally monotone operatorsprojected evolutionary liftingwell-posednessinfinite-horizon problems

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The pith

Projected evolutionary lifting establishes existence and uniqueness of mild solutions to stationary HJB equations in separable Hilbert spaces for every discount rate greater than zero.

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

The paper proves that stationary Hamilton-Jacobi-Bellman equations linked to infinite-horizon stochastic optimal control problems in infinite-dimensional spaces admit unique mild solutions even when the transition semigroup has no global smoothing, control operators are unbounded and singular, and running costs depend on the state. The key step is to apply a projected evolutionary lifting that enlarges the state space and recasts the equation in a form where the theory of maximally monotone operators directly yields the result. This removes the earlier restriction to sufficiently large discount factors that arose from contraction-mapping arguments, so the well-posedness statement now holds for arbitrary positive discount rates.

Core claim

The projected evolutionary lifting transforms the original stationary HJB equation into an equivalent problem in an extended space to which the theory of maximally monotone operators applies, thereby proving existence and uniqueness of mild solutions for any discount factor λ > 0.

What carries the argument

Projected Evolutionary Lifting, a technique that enlarges the state space by incorporating an evolutionary variable and suitable projections so that the HJB equation becomes amenable to monotone-operator methods without requiring global smoothing of the semigroup.

If this is right

Optimal control problems in infinite dimensions can now be characterized by unique mild solutions of the stationary HJB equation without restricting the discount rate.
The same lifting procedure applies directly to problems whose transition semigroup lacks smoothing and whose control operators are singular.
Value functions arising from infinite-horizon problems become well-defined objects that can be used to construct optimal feedback controls in the original Hilbert space.
The framework accommodates state-dependent running costs without additional regularity assumptions on the cost functional.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method suggests that similar liftings could be used for time-dependent HJB equations or for control problems driven by other infinite-dimensional noise processes.
Finite-dimensional Galerkin approximations of the lifted equation might converge to the infinite-dimensional mild solution, providing a route to numerical schemes.
The monotone-operator reformulation may allow direct comparison with viscosity-solution approaches in the same infinite-dimensional setting.

Load-bearing premise

The lifting must convert the original dynamics and cost into a setting where the associated operator is maximally monotone and the fixed-point or resolvent equation admits a unique solution.

What would settle it

Exhibit a concrete stochastic control problem in a separable Hilbert space, with unbounded control operator and state-dependent cost, for which the lifted stationary equation possesses either no mild solution or more than one mild solution when λ > 0 is arbitrary.

read the original abstract

This paper establishes the existence and uniqueness of mild solutions to stationary Hamilton-Jacobi-Bellman (HJB) equations associated with infinite-horizon stochastic optimal control problems in separable Hilbert spaces. Our framework includes settings with a lack of global smoothing properties of the transition semigroup, singular dynamics involving unbounded control operators, and state-dependent running costs. We overcome these challenges by lifting the state space using the Projected Evolutionary Lifting technique. This work is an extension of G. Bolli and F. Gozzi, Lifting and partial smoothing for stationary HJB equations and related control problems in infinite dimensions, 2025, in which existence and uniqueness is proved via a contraction mapping argument and is consequently restricted to sufficiently large discount factors. We remove this restriction, proving existence and uniqueness for any discount rate $\lambda > 0$ using tools from the theory of maximally monotone operators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Bolli and Fuchs drop the large-discount limit from their 2025 HJB result by lifting the problem into a monotone-operator setting that works for any lambda > 0.

read the letter

The main takeaway is that this paper removes the restriction to sufficiently large discount factors that limited their earlier contraction-mapping proof. By introducing projected evolutionary lifting, they recast the stationary HJB equation so that standard results on maximal monotone operators deliver existence and uniqueness of mild solutions for arbitrary lambda > 0, even when the transition semigroup lacks global smoothing, the control operator is unbounded, and the running cost depends on the state. That extension is the concrete advance over the 2025 work. The approach stays within familiar functional-analytic territory—semigroup theory plus Minty-Browder surjectivity—so the tools themselves are not exotic. The lifting construction is presented as a device that absorbs the singular parts without redefining the original problem, which keeps the argument reasonably direct. The framework covers a useful range of infinite-dimensional stochastic control problems that previously sat outside the contraction regime. The soft spot is the verification that the lifted operator actually satisfies the range condition once the projection is applied. With unbounded controls and state-dependent costs, monotonicity may carry over, but coercivity or surjectivity is not automatic from the stated assumptions on the semigroup and cost; the earlier contraction argument simply avoided this check by restricting lambda. The abstract supplies no explicit estimates or assumption checks, so it is not yet clear whether the projection step closes the gap or leaves a hole. A reader working in infinite-dimensional optimal control or PDE-constrained optimization would get value from seeing how the lifting is built and whether it generalizes. The paper shows clear engagement with the literature and a substantive claim, so it deserves a serious referee even if revisions will be needed to document the range-condition argument in detail. I would send it out for review rather than desk-reject.

Referee Report

1 major / 2 minor

Summary. The paper claims to establish existence and uniqueness of mild solutions to stationary Hamilton-Jacobi-Bellman equations for infinite-horizon stochastic optimal control problems in separable Hilbert spaces. It introduces the Projected Evolutionary Lifting to handle transition semigroups lacking global smoothing, singular unbounded control operators, and state-dependent running costs, extending prior contraction-mapping results (limited to large discount factors) by applying maximally monotone operator theory to obtain well-posedness for arbitrary discount rates λ > 0.

Significance. If the lifting construction is shown to produce a maximally monotone operator satisfying the range condition, the result would be significant for infinite-dimensional stochastic control theory. It removes the large-λ restriction of contraction-mapping approaches, broadening applicability to problems with singular dynamics and non-quadratic costs while relying on standard semigroup and monotone-operator tools in a novel way.

major comments (1)

[Construction of the lifted operator and application of monotone operator theory (around the definition of A_λ and the H_] The central claim rests on the projected evolutionary lifting producing a maximally monotone operator A_λ satisfying the range condition R(λI + A_λ) = X for arbitrary λ > 0. The assumptions on the transition semigroup (no global smoothing), unbounded control operator B, and state-dependent running cost do not automatically guarantee this coercivity or surjectivity in the lifted space; an explicit verification or additional hypothesis is needed to apply the Minty-Browder theorem beyond the large-λ regime of the prior contraction argument.

minor comments (2)

[Introduction and preliminaries] Clarify the precise definition of 'mild solution' in the infinite-dimensional setting early in the paper, including how it relates to the lifted space.
[Introduction] The reference to Bolli and Gozzi (2025) should include a brief comparison of the contraction-mapping limitation versus the new monotone-operator approach.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below, providing clarification on the construction and properties of the lifted operator while offering to improve the exposition.

read point-by-point responses

Referee: The central claim rests on the projected evolutionary lifting producing a maximally monotone operator A_λ satisfying the range condition R(λI + A_λ) = X for arbitrary λ > 0. The assumptions on the transition semigroup (no global smoothing), unbounded control operator B, and state-dependent running cost do not automatically guarantee this coercivity or surjectivity in the lifted space; an explicit verification or additional hypothesis is needed to apply the Minty-Browder theorem beyond the large-λ regime of the prior contraction argument.

Authors: We appreciate the referee drawing attention to this foundational step. The projected evolutionary lifting is constructed precisely so that A_λ is maximally monotone on the lifted space for every λ > 0, with the range condition following from a coercivity estimate that exploits the projection onto the graph of the semigroup evolution. Monotonicity is verified directly via the inner product in the product space, using dissipativity of the transition semigroup and the form of the running cost; the lack of global smoothing is compensated by the projection, while the unbounded control operator B is accommodated through the domain of the generator in the lifted operator. The range condition R(λI + A_λ) = X is obtained by solving the associated resolvent equation, which reduces to a well-posed fixed-point problem in the lifted space that holds uniformly for all λ > 0, thereby removing the large-λ restriction of the earlier contraction-mapping approach. We do not believe an additional hypothesis is required under the stated assumptions, but we agree that the verification can be made more explicit. We will revise the manuscript to include a dedicated lemma that isolates the application of the Minty-Browder theorem with expanded intermediate steps. revision: partial

Circularity Check

0 steps flagged

No significant circularity; the existence result for arbitrary discount rates applies external monotone operator theory to the projected lifting, independent of the prior contraction-mapping result.

full rationale

The derivation chain begins from the HJB equation under the stated assumptions on the semigroup, unbounded control operator B, and state-dependent costs. The Projected Evolutionary Lifting is introduced to recast the problem so that the resulting operator satisfies the conditions for Minty-Browder surjectivity (maximal monotonicity plus range condition). This step invokes standard results from monotone operator theory in Hilbert spaces, which are external and not derived within the paper. The self-citation to Bolli-Gozzi 2025 is limited to contrasting the previous large-λ contraction argument; the current proof does not invoke or rely on that contraction result or any uniqueness theorem from the overlapping-author prior work to establish the new claim for all λ>0. No equation is shown to equal its input by definition, no fitted parameter is relabeled as a prediction, and no ansatz or uniqueness is smuggled via self-citation. The framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on standard background from semigroup theory and monotone operators plus the validity of the newly introduced lifting map; no numerical fitting occurs.

axioms (2)

domain assumption Transition semigroup exists on separable Hilbert space without global smoothing
Invoked to describe the class of dynamics the lifting must handle.
standard math Maximally monotone operator theory yields mild solutions under the lifted dynamics
Core tool used to obtain existence and uniqueness for any lambda > 0.

invented entities (1)

Projected Evolutionary Lifting no independent evidence
purpose: Embeds the original state space into a larger space to restore sufficient regularity for monotone operator arguments
New construction introduced to overcome lack of smoothing and unbounded controls; no independent falsifiable prediction is given in the abstract.

pith-pipeline@v0.9.0 · 5447 in / 1351 out tokens · 82081 ms · 2026-05-08T17:17:45.516492+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Barbu,Nonlinear differential equations of monotone types in Banach spaces(Springer Monographs in Mathematics)

V. Barbu,Nonlinear differential equations of monotone types in Banach spaces(Springer Monographs in Mathematics). Springer, New York, 2010, pp. x+272

work page 2010
[2]

Bolli and F

G. Bolli and F. de Feo,Optimal control of stochastic volterra integral equations with completely monotone kernels and stochastic differential equations on hilbert spaces with unbounded control and diffusion operators, 2026. arXiv:2602.17578 [math.OC]

work page arXiv 2026
[3]

Bolli and F

G. Bolli and F. Gozzi,Lifting and partial smoothing for stationary HJB equations and related control problems in infinite dimensions, 2025. arXiv:2510.25894 [math.OC]

work page arXiv 2025
[4]

Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces

P. Cannarsa and G. Da Prato, “Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces”, English, inStochastic partial differential equations and applications. Proceedings of the third meeting on stochastic partial differential equations and applications held at Villa Madruzzo, Trento, Italy, January 1990, Harlow: Longman Scientific & ...

work page 1990
[5]

Second-order Hamilton-Jacobi equations in infinite dimen- sions

P. Cannarsa and G. da Prato, “Second-order Hamilton-Jacobi equations in infinite dimen- sions”, English,SIAM J. Control Optim., vol. 29, no. 2, pp. 474–492, 1991

work page 1991
[6]

Infinite-dimensional Hamilton-Jacobi-Bellman equations in Gauss-Sobolev spaces

P.-L. Chow and J. -L. Menaldi, “Infinite-dimensional Hamilton-Jacobi-Bellman equations in Gauss-Sobolev spaces”, English,Nonlinear Anal., Theory Methods Appl., vol. 29, no. 4, pp. 415–426, 1997

work page 1997
[7]

Da Prato and J

G. Da Prato and J. Zabczyk,Second order partial differential equations in Hilbert spaces (London Mathematical Society Lecture Note Series). Cambridge University Press, Cam- bridge, 2002, vol. 293, pp. xvi+379

work page 2002
[8]

Da Prato and J

G. Da Prato and J. Zabczyk,Stochastic equations in infinite dimensions(Encyclopedia of Mathematics and its Applications), Second. Cambridge University Press, Cambridge, 2014, vol. 152, pp. xviii+493

work page 2014
[9]

Fabbri, F

G. Fabbri, F. Gozzi, and A. Swiech,Stochastic optimal control in infinite dimension(Prob- ability Theory and Stochastic Modelling). Springer, Cham, 2017, vol. 82, pp. xxiii+916, Dynamic programming and HJB equations, With a contribution by Marco Fuhrman and Gianmario Tessitore

work page 2017
[10]

Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces

M. Fuhrman and G. Tessitore, “Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces.”, English,Ann. Probab., vol. 32, no. 1B, pp. 607– 660, 2004

work page 2004
[11]

Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control

M. Fuhrman and G. Tessitore, “Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control”, English,Ann. Probab., vol. 30, no. 3, pp. 1397–1465, 2002

work page 2002
[12]

The Bismut-Elworthy formula for backward SDE’s and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces

M. Fuhrman and G. Tessitore, “The Bismut-Elworthy formula for backward SDE’s and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces”, English,Stochastics Stochastics Rep., vol. 74, no. 1-2, pp. 429–464, 2002

work page 2002
[13]

Second order parabolic Hamilton-Jacobi-Bellman equations in Hilbert spaces and stochastic control: L2 µ approach

B. Goldys and F. Gozzi, “Second order parabolic Hamilton-Jacobi-Bellman equations in Hilbert spaces and stochastic control: L2 µ approach”, English,Stochastic Processes Appl., vol. 116, no. 12, pp. 1932–1963, 2006

work page 1932
[14]

Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities

F. Gozzi, “Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities”, English,J. Math. Anal. Appl., vol. 198, no. 2, pp. 399–443, 1996

work page 1996
[15]

Lifting partial smoothing to solve HJB equations and stochastic control problems

F. Gozzi and F. Masiero, “Lifting partial smoothing to solve HJB equations and stochastic control problems”,SIAM J. Control Optim., vol. 63, no. 3, pp. 1515–1559, 2025

work page 2025
[16]

Stochastic control problems with unbounded control operators: Solutions through generalized derivatives

F. Gozzi and F. Masiero, “Stochastic control problems with unbounded control operators: Solutions through generalized derivatives”,SIAM J. Control Optim., vol. 61, no. 2, pp. 586– 619, 2023

work page 2023
[17]

Stochastic optimal control with delay in the control I: Solving the HJB equation through partial smoothing

F. Gozzi and F. Masiero, “Stochastic optimal control with delay in the control I: Solving the HJB equation through partial smoothing”,SIAM J. Control Optim., vol. 55, no. 5, pp. 2981–3012, 2017. REFERENCES 19

work page 2017
[18]

Lasiecka and R

I. Lasiecka and R. Triggiani,Control theory for partial differential equations: continuous and approximation theories. I(Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge, 2000, vol. 74, pp. xxii+644+I4, Abstract parabolic systems

work page 2000
[19]

Lions and E

J.-L. Lions and E. Magenes,Non-homogeneous boundary value problems and applications. Vol. I(Die Grundlehren der mathematischen Wissenschaften). Springer-Verlag, New York-Heidelberg, 1972, vol. Band 181, pp. xvi+357, Translated from the French by P. Kenneth

work page 1972
[20]

Partial smoothing of delay transition semigroups acting on special functions

F. Masiero and G. Tessitore, “Partial smoothing of delay transition semigroups acting on special functions”, English,J. Differ. Equations, vol. 316, pp. 599–640, 2022

work page 2022
[21]

Optimal boundary control of reaction–diffusion partial differential equations via weak variations

S. J. Moura and H. K. Fathy, “Optimal boundary control of reaction–diffusion partial differential equations via weak variations”,Journal of Dynamic Systems, Measurement, and Control, vol. 135, no. 3, no. 034501, Feb. 2013

work page 2013
[22]

Pazy,Semigroups of linear operators and applications to partial differential equations (Applied Mathematical Sciences)

A. Pazy,Semigroups of linear operators and applications to partial differential equations (Applied Mathematical Sciences). Springer-Verlag, New York, 1983, vol. 44, pp. viii+279

work page 1983
[23]

Yosida,Functional analysis(Die Grundlehren der mathematischen Wissenschaften)

K. Yosida,Functional analysis(Die Grundlehren der mathematischen Wissenschaften). Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965, vol. Band 123, pp. xi+458. Dipartimento di Matematica Guido Castelnuovo, Universit `a degli Studi di Roma La Sapienza, Roma, Italy Email address:gabriele.bolli@uniroma1.it Department of AI, Data and Decision Scie...

work page 1965

[1] [1]

Barbu,Nonlinear differential equations of monotone types in Banach spaces(Springer Monographs in Mathematics)

V. Barbu,Nonlinear differential equations of monotone types in Banach spaces(Springer Monographs in Mathematics). Springer, New York, 2010, pp. x+272

work page 2010

[2] [2]

Bolli and F

G. Bolli and F. de Feo,Optimal control of stochastic volterra integral equations with completely monotone kernels and stochastic differential equations on hilbert spaces with unbounded control and diffusion operators, 2026. arXiv:2602.17578 [math.OC]

work page arXiv 2026

[3] [3]

Bolli and F

G. Bolli and F. Gozzi,Lifting and partial smoothing for stationary HJB equations and related control problems in infinite dimensions, 2025. arXiv:2510.25894 [math.OC]

work page arXiv 2025

[4] [4]

Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces

P. Cannarsa and G. Da Prato, “Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces”, English, inStochastic partial differential equations and applications. Proceedings of the third meeting on stochastic partial differential equations and applications held at Villa Madruzzo, Trento, Italy, January 1990, Harlow: Longman Scientific & ...

work page 1990

[5] [5]

Second-order Hamilton-Jacobi equations in infinite dimen- sions

P. Cannarsa and G. da Prato, “Second-order Hamilton-Jacobi equations in infinite dimen- sions”, English,SIAM J. Control Optim., vol. 29, no. 2, pp. 474–492, 1991

work page 1991

[6] [6]

Infinite-dimensional Hamilton-Jacobi-Bellman equations in Gauss-Sobolev spaces

P.-L. Chow and J. -L. Menaldi, “Infinite-dimensional Hamilton-Jacobi-Bellman equations in Gauss-Sobolev spaces”, English,Nonlinear Anal., Theory Methods Appl., vol. 29, no. 4, pp. 415–426, 1997

work page 1997

[7] [7]

Da Prato and J

G. Da Prato and J. Zabczyk,Second order partial differential equations in Hilbert spaces (London Mathematical Society Lecture Note Series). Cambridge University Press, Cam- bridge, 2002, vol. 293, pp. xvi+379

work page 2002

[8] [8]

Da Prato and J

G. Da Prato and J. Zabczyk,Stochastic equations in infinite dimensions(Encyclopedia of Mathematics and its Applications), Second. Cambridge University Press, Cambridge, 2014, vol. 152, pp. xviii+493

work page 2014

[9] [9]

Fabbri, F

G. Fabbri, F. Gozzi, and A. Swiech,Stochastic optimal control in infinite dimension(Prob- ability Theory and Stochastic Modelling). Springer, Cham, 2017, vol. 82, pp. xxiii+916, Dynamic programming and HJB equations, With a contribution by Marco Fuhrman and Gianmario Tessitore

work page 2017

[10] [10]

Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces

M. Fuhrman and G. Tessitore, “Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces.”, English,Ann. Probab., vol. 32, no. 1B, pp. 607– 660, 2004

work page 2004

[11] [11]

Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control

M. Fuhrman and G. Tessitore, “Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control”, English,Ann. Probab., vol. 30, no. 3, pp. 1397–1465, 2002

work page 2002

[12] [12]

The Bismut-Elworthy formula for backward SDE’s and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces

M. Fuhrman and G. Tessitore, “The Bismut-Elworthy formula for backward SDE’s and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces”, English,Stochastics Stochastics Rep., vol. 74, no. 1-2, pp. 429–464, 2002

work page 2002

[13] [13]

Second order parabolic Hamilton-Jacobi-Bellman equations in Hilbert spaces and stochastic control: L2 µ approach

B. Goldys and F. Gozzi, “Second order parabolic Hamilton-Jacobi-Bellman equations in Hilbert spaces and stochastic control: L2 µ approach”, English,Stochastic Processes Appl., vol. 116, no. 12, pp. 1932–1963, 2006

work page 1932

[14] [14]

Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities

F. Gozzi, “Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities”, English,J. Math. Anal. Appl., vol. 198, no. 2, pp. 399–443, 1996

work page 1996

[15] [15]

Lifting partial smoothing to solve HJB equations and stochastic control problems

F. Gozzi and F. Masiero, “Lifting partial smoothing to solve HJB equations and stochastic control problems”,SIAM J. Control Optim., vol. 63, no. 3, pp. 1515–1559, 2025

work page 2025

[16] [16]

Stochastic control problems with unbounded control operators: Solutions through generalized derivatives

F. Gozzi and F. Masiero, “Stochastic control problems with unbounded control operators: Solutions through generalized derivatives”,SIAM J. Control Optim., vol. 61, no. 2, pp. 586– 619, 2023

work page 2023

[17] [17]

Stochastic optimal control with delay in the control I: Solving the HJB equation through partial smoothing

F. Gozzi and F. Masiero, “Stochastic optimal control with delay in the control I: Solving the HJB equation through partial smoothing”,SIAM J. Control Optim., vol. 55, no. 5, pp. 2981–3012, 2017. REFERENCES 19

work page 2017

[18] [18]

Lasiecka and R

I. Lasiecka and R. Triggiani,Control theory for partial differential equations: continuous and approximation theories. I(Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge, 2000, vol. 74, pp. xxii+644+I4, Abstract parabolic systems

work page 2000

[19] [19]

Lions and E

J.-L. Lions and E. Magenes,Non-homogeneous boundary value problems and applications. Vol. I(Die Grundlehren der mathematischen Wissenschaften). Springer-Verlag, New York-Heidelberg, 1972, vol. Band 181, pp. xvi+357, Translated from the French by P. Kenneth

work page 1972

[20] [20]

Partial smoothing of delay transition semigroups acting on special functions

F. Masiero and G. Tessitore, “Partial smoothing of delay transition semigroups acting on special functions”, English,J. Differ. Equations, vol. 316, pp. 599–640, 2022

work page 2022

[21] [21]

Optimal boundary control of reaction–diffusion partial differential equations via weak variations

S. J. Moura and H. K. Fathy, “Optimal boundary control of reaction–diffusion partial differential equations via weak variations”,Journal of Dynamic Systems, Measurement, and Control, vol. 135, no. 3, no. 034501, Feb. 2013

work page 2013

[22] [22]

Pazy,Semigroups of linear operators and applications to partial differential equations (Applied Mathematical Sciences)

A. Pazy,Semigroups of linear operators and applications to partial differential equations (Applied Mathematical Sciences). Springer-Verlag, New York, 1983, vol. 44, pp. viii+279

work page 1983

[23] [23]

Yosida,Functional analysis(Die Grundlehren der mathematischen Wissenschaften)

K. Yosida,Functional analysis(Die Grundlehren der mathematischen Wissenschaften). Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965, vol. Band 123, pp. xi+458. Dipartimento di Matematica Guido Castelnuovo, Universit `a degli Studi di Roma La Sapienza, Roma, Italy Email address:gabriele.bolli@uniroma1.it Department of AI, Data and Decision Scie...

work page 1965