Introduction In-vivo control - Theory In-vivo control - Example Conclusion
A Control Theory for Stochastic Biomolecular Regulation
Corentin Briat, Ankit Gupta and Mustafa Khammash
SIAM Conference on Control and its Applications - 08/07/2015
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 0/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Introduction
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 0/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Reaction networks
A reaction network is. . .• A set of d distinct species X1, . . . ,Xd
• A set of K reactions R1, . . . , RK specifying how species interact with each otherand for each reaction we have
• A stoichiometric vector ζk ∈ Zd describing how reactions change the state value• A propensity function λk ∈ R≥0 describing the "strength" of the reaction
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 1/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Reaction networks
A reaction network is. . .• A set of d distinct species X1, . . . ,Xd
• A set of K reactions R1, . . . , RK specifying how species interact with each otherand for each reaction we have
• A stoichiometric vector ζk ∈ Zd describing how reactions change the state value• A propensity function λk ∈ R≥0 describing the "strength" of the reaction
Deterministic networks• Large populations (concentrations are well-defined), e.g. as in chemistry• Lots of analytical tools, e.g. reaction network theory, dynamical systems theory,
Lyapunov theory of stability, nonlinear control theory, etc.
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 1/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Reaction networks
A reaction network is. . .• A set of d distinct species X1, . . . ,Xd
• A set of K reactions R1, . . . , RK specifying how species interact with each otherand for each reaction we have
• A stoichiometric vector ζk ∈ Zd describing how reactions change the state value• A propensity function λk ∈ R≥0 describing the "strength" of the reaction
Deterministic networks• Large populations (concentrations are well-defined), e.g. as in chemistry• Lots of analytical tools, e.g. reaction network theory, dynamical systems theory,
Lyapunov theory of stability, nonlinear control theory, etc.
Stochastic networks• Low populations (concentrations are NOT well defined)• Biological processes where key molecules are in low copy number (mRNA '10
copies per cell)• No well-established theory for biology, “analysis" often based on simulations. . .• No well-established control theory
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 1/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Chemical master equation
State and dynamics
• The state X ∈ Nd0 is vector of random variables representing molecules count• The dynamics of the process is described by a jump Markov process (X(t))t≥0
Chemical Master Equation (Forward Kolmogorov equation)
px0 (x, t) =K∑k=1
λk(x− ζk)px0 (x− ζk, t)− λk(x)px0 (x, t), x ∈ Nd0
where px0 (x, t) = P[X(t) = x|X(0) = x0] and px0 (x, 0) = δx0 (x).
Solving the CME
• Infinite countable number of linear time-invariant ODEs• Exactly solvable only in very simple cases• Some numerical schemes are available (FSP, QTT, etc) but limited by the curse of
dimensionality; if X ∈ {0, . . . , x− 1}d, then we have xd states
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Chemical master equation
State and dynamics
• The state X ∈ Nd0 is vector of random variables representing molecules count• The dynamics of the process is described by a jump Markov process (X(t))t≥0
Chemical Master Equation (Forward Kolmogorov equation)
px0 (x, t) =K∑k=1
λk(x− ζk)px0 (x− ζk, t)− λk(x)px0 (x, t), x ∈ Nd0
where px0 (x, t) = P[X(t) = x|X(0) = x0] and px0 (x, 0) = δx0 (x).
Solving the CME
• Infinite countable number of linear time-invariant ODEs• Exactly solvable only in very simple cases• Some numerical schemes are available (FSP, QTT, etc) but limited by the curse of
dimensionality; if X ∈ {0, . . . , x− 1}d, then we have xd states
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 2/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Ergodicity of reaction networks
ErgodicityA given stochastic reaction network is ergodic if there is a probability distribution πsuch that for all x0 ∈ Nd0 , we have that px0 (x, t)→ π as t→∞.
Theorem (Condition for ergodicity1)Assume that
(a) the state-space of the network is irreducible; and
(b) there exists a norm-like function V (x) such that the drift conditionK∑i=1
λi(x)[V (x+ ζi)− V (x)] ≤ c1 − c2V (x)
holds for some c1, c2 > 0 and for all x ∈ Nd0 .Then, the stochastic reaction networkis ergodic.
Choosing V (x) = 〈v, x〉, v > 0, allows to establish the ergodicity of a wide class ofexisting reaction networks2
1 S. P. Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. , 19932 A. Gupta, C. Briat, and M. Khammash. A scalable computational framework for establishing long-term behavior of stochastic reaction networks,
PLOS Computational Biology, 2014
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 3/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Ergodicity of reaction networks
ErgodicityA given stochastic reaction network is ergodic if there is a probability distribution πsuch that for all x0 ∈ Nd0 , we have that px0 (x, t)→ π as t→∞.
Theorem (Condition for ergodicity1)Assume that
(a) the state-space of the network is irreducible; and
(b) there exists a norm-like function V (x) such that the drift conditionK∑i=1
λi(x)[V (x+ ζi)− V (x)] ≤ c1 − c2V (x)
holds for some c1, c2 > 0 and for all x ∈ Nd0 .Then, the stochastic reaction networkis ergodic.
Choosing V (x) = 〈v, x〉, v > 0, allows to establish the ergodicity of a wide class ofexisting reaction networks2
1 S. P. Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. , 19932 A. Gupta, C. Briat, and M. Khammash. A scalable computational framework for establishing long-term behavior of stochastic reaction networks,
PLOS Computational Biology, 2014
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 3/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Ergodicity of reaction networks
ErgodicityA given stochastic reaction network is ergodic if there is a probability distribution πsuch that for all x0 ∈ Nd0 , we have that px0 (x, t)→ π as t→∞.
Theorem (Condition for ergodicity1)Assume that
(a) the state-space of the network is irreducible; and
(b) there exists a norm-like function V (x) such that the drift conditionK∑i=1
λi(x)[V (x+ ζi)− V (x)] ≤ c1 − c2V (x)
holds for some c1, c2 > 0 and for all x ∈ Nd0 .Then, the stochastic reaction networkis ergodic.
Choosing V (x) = 〈v, x〉, v > 0, allows to establish the ergodicity of a wide class ofexisting reaction networks2
1 S. P. Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. , 19932 A. Gupta, C. Briat, and M. Khammash. A scalable computational framework for establishing long-term behavior of stochastic reaction networks,
PLOS Computational Biology, 2014
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 3/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Control problems
In-silico control• Controllers are implemented outside cells• Single cell1 or population control2
In-vivo control• Controllers are implemented inside cells• Single cell and population control3
1 J. Uhlendorf, et al. Long-term model predictive control of gene expression at the population and single-cell levels, Proceedings of the National
Academy of Sciences of the United States of America, 20122 A. Milias-Argeitis, et al. In silico feedback for in vivo regulation of a gene expression circuit, Nature Biotechnology, 20113 C. Briat, A. Gupta, and M. Khammash. Integral feedback generically achieves perfect adaptation in stochastic biochemical networks, ArXiV, 2015
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 4/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Control problems
In-silico control• Controllers are implemented outside cells• Single cell1 or population control2
In-vivo control• Controllers are implemented inside cells• Single cell and population control3
1 J. Uhlendorf, et al. Long-term model predictive control of gene expression at the population and single-cell levels, Proceedings of the National
Academy of Sciences of the United States of America, 20122 A. Milias-Argeitis, et al. In silico feedback for in vivo regulation of a gene expression circuit, Nature Biotechnology, 20113 C. Briat, A. Gupta, and M. Khammash. Integral feedback generically achieves perfect adaptation in stochastic biochemical networks, ArXiV, 2015
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 4/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
In-vivo population control - Theory
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 4/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Setup
Open-loop reaction network
• d molecular species: X1, . . . ,Xd
• X1 is the actuated species: ∅ u−−−→X1
• Measured/controlled species: Y = X`
Problem
Find a controller such that the closed-loop network is ergodic and such that we haveE[Y (t)]→ µ as t→∞, for some reference value µ
The controller• Two species Z1 and Z2.
∅ µ−−−→ Z1︸ ︷︷ ︸reference
, ∅ Y−−−→ Z2︸ ︷︷ ︸measurement
, Z1 + Z2η−−−→ ∅︸ ︷︷ ︸
comparison
, ∅ kZ1−−−→X1︸ ︷︷ ︸actuation
.
where k, η > 0 are control parameters.
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 5/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Setup
Open-loop reaction network
• d molecular species: X1, . . . ,Xd
• X1 is the actuated species: ∅ u−−−→X1
• Measured/controlled species: Y = X`
Problem
Find a controller such that the closed-loop network is ergodic and such that we haveE[Y (t)]→ µ as t→∞, for some reference value µ
The controller• Two species Z1 and Z2.
∅ µ−−−→ Z1︸ ︷︷ ︸reference
, ∅ Y−−−→ Z2︸ ︷︷ ︸measurement
, Z1 + Z2η−−−→ ∅︸ ︷︷ ︸
comparison
, ∅ kZ1−−−→X1︸ ︷︷ ︸actuation
.
where k, η > 0 are control parameters.
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 5/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Setup
Open-loop reaction network
• d molecular species: X1, . . . ,Xd
• X1 is the actuated species: ∅ u−−−→X1
• Measured/controlled species: Y = X`
Problem
Find a controller such that the closed-loop network is ergodic and such that we haveE[Y (t)]→ µ as t→∞, for some reference value µ
The controller• Two species Z1 and Z2.
∅ µ−−−→ Z1︸ ︷︷ ︸reference
, ∅ Y−−−→ Z2︸ ︷︷ ︸measurement
, Z1 + Z2η−−−→ ∅︸ ︷︷ ︸
comparison
, ∅ kZ1−−−→X1︸ ︷︷ ︸actuation
.
where k, η > 0 are control parameters.
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 5/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
The hidden integral action1
Moments equations
d
dtE[Z1(t)] = µ− ηE[Z1(t)Z2(t)]
d
dtE[Z2(t)] = E[Y (t)]− ηE[Z1(t)Z2(t)].
Integral action
• We have thatd
dtE[Z1(t)− Z2(t)] = µ− E[Y (t)],
so we have an integral action on the mean• Closed-loop ergodic⇒ E[Y (t)]→ µ as t→∞• No need for solving moments equations→ no closure problem :)
1 K. Oishi and E. Klavins. Biomolecular implementation of linear I/O systems, IET Systems Biology, 2010
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
The hidden integral action1
Moments equations
d
dtE[Z1(t)] = µ− ηE[Z1(t)Z2(t)]
d
dtE[Z2(t)] = E[Y (t)]− ηE[Z1(t)Z2(t)].
Integral action
• We have thatd
dtE[Z1(t)− Z2(t)] = µ− E[Y (t)],
so we have an integral action on the mean• Closed-loop ergodic⇒ E[Y (t)]→ µ as t→∞• No need for solving moments equations→ no closure problem :)
1 K. Oishi and E. Klavins. Biomolecular implementation of linear I/O systems, IET Systems Biology, 2010
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
General stabilization result
TheoremLet V (x) = 〈v, x〉 with v ∈ Rd>0 and W (x) = 〈w, x〉 with w ∈ Rd≥0, w1, w` > 0.Assume that
(a) the state-space of the open-loop reaction network is irreducible; and
(b) there exist c1, c3 > 0 and c2 ≥ 0 such that
K∑k=1
λk(x)[V (x+ ζk)− V (x)] ≤ −c1V (x),
K∑k=1
λk(x)[W (x+ ζk)−W (x)] ≥ −c2 − c3x`,(1)
hold for all x ∈ Nd0 (together with some other technical conditions).
Then, the closed-loop network is ergodic and we have that E[Y (t)]→ µ as t→∞.
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Unimolecular networks
TheoremLet us consider a unimolecular reaction network with irreducible state-space. Assumethat its first-order moments system
d
dtE[X(t)] = AE[X(t)] + e1u(t)
y(t) = eT` E[X(t)](2)
is
(a) asymptotically stable, i.e A Hurwitz stable (LP)
(b) output controllable, i.e. rank[eT` e1 eT` Ae1 . . . eT` A
d−1e1]
= 1 (LP)
Then, for all control parameters k, η > 0,
(a) the closed-loop reaction network (system + controller) is ergodic
(b) all the first and second order moments of the random variables X1, . . . , Xd areuniformly bounded and globally converging
(c) E[Y (t)]→ µ as t→∞.
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Unimolecular networks
TheoremLet us consider a unimolecular reaction network with irreducible state-space. Assumethat its first-order moments system
d
dtE[X(t)] = AE[X(t)] + e1u(t)
y(t) = eT` E[X(t)](2)
is
(a) asymptotically stable, i.e A Hurwitz stable (LP)
(b) output controllable, i.e. rank[eT` e1 eT` Ae1 . . . eT` A
d−1e1]
= 1 (LP)
Then, for all control parameters k, η > 0,
(a) the closed-loop reaction network (system + controller) is ergodic
(b) all the first and second order moments of the random variables X1, . . . , Xd areuniformly bounded and globally converging
(c) E[Y (t)]→ µ as t→∞.
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 8/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Unimolecular networks
TheoremLet us consider a unimolecular reaction network with irreducible state-space. Assumethat its first-order moments system
d
dtE[X(t)] = AE[X(t)] + e1u(t)
y(t) = eT` E[X(t)](2)
is
(a) asymptotically stable, i.e A Hurwitz stable (LP)
(b) output controllable, i.e. rank[eT` e1 eT` Ae1 . . . eT` A
d−1e1]
= 1 (LP)
Then, for all control parameters k, η > 0,
(a) the closed-loop reaction network (system + controller) is ergodic
(b) all the first and second order moments of the random variables X1, . . . , Xd areuniformly bounded and globally converging
(c) E[Y (t)]→ µ as t→∞.
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Properties
Closed-loop system
• Robust ergodicity, tracking and disturbance rejection• Population control is achieved
Controller• Innocuous: open-loop ergodic & output controllable⇒ closed-loop ergodic• Decentralized: use only local information (single-cell control)• Implementable: small number of reactions
Additional remarks• No moment closure problem• Expected to work on a wide class of networks (even though the theory is not there
yet)
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
In-vivo population control - Example
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Gene expression network d = 2, K = 4
R1 : ∅ kr−−−→ mRNA (X1)
R2 : mRNAγr−−−→ ∅
R3 : mRNAkp−−−→ mRNA+protein (X1 + X2)
R4 : proteinγp−−−→ ∅
S =[ζ1 ζ2 ζ3 ζ4
]λ(x) = [ λ1(x) λ2(x) λ3(x) λ4(x) ]T
=
[1 −1 0 00 0 1 −1
]= [ kr γrx1 kpx1 γpx2 ]T
We want to control the average number of proteins by suitably acting on thetranscription rate kr
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 10/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Gene expression network d = 2, K = 4
R1 : ∅ kr−−−→ mRNA (X1)
R2 : mRNAγr−−−→ ∅
R3 : mRNAkp−−−→ mRNA+protein (X1 + X2)
R4 : proteinγp−−−→ ∅
S =[ζ1 ζ2 ζ3 ζ4
]λ(x) = [ λ1(x) λ2(x) λ3(x) λ4(x) ]T
=
[1 −1 0 00 0 1 −1
]= [ kr γrx1 kpx1 γpx2 ]T
We want to control the average number of proteins by suitably acting on thetranscription rate kr
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 10/14
Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Gene expression control
TheoremFor any values of the system parameters kp, γr, γp > 0 and the control parametersµ, k, η > 0, the closed-loop network is ergodic and we have that E[X2(t)]→ µ ast→∞ globally.
0 10 20 30 40 50 60 700
2
4
6
8
10
12
14
16
18
Time t
Pop
ulation
[Molecules]
X1(t)X2(t)Z1(t)Z2(t)
0 10 20 30 40 50 60 700
1
2
3
4
5
6
7
8
9
10
Time
Pop
ulation
averages
[Molecules]
E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]
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Comparison with deterministic control
Deterministic
x1 = kz1 − γrx1x2 = kpx1 − γpx2z1 = µ− ηz1z2z2 = x2 − ηz1z2
0 5 10 15 20 25 300
1
2
3
4
5
6
7
Time
Pop
ulation
concentrations
x1(t)x2(t)z1(t)z2(t)
Stochastic
E[X1] = kE[Z1]− γrE[X1]
E[X2] = kpE[X1]− γpE[X2]
E[Z1] = µ− ηE[Z1]E[Z2]−ηV (Z1, Z2)
E[Z2] = E[X2]− ηE[Z1]E[Z2]−ηV (Z1, Z2)
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
Time
Pop
ulation
averages
[Molecules]
E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Comparison with deterministic control
Deterministic
x1 = kz1 − γrx1x2 = kpx1 − γpx2z1 = µ− ηz1z2z2 = x2 − ηz1z2
0 5 10 15 20 25 300
1
2
3
4
5
6
7
Time
Pop
ulation
concentrations
x1(t)x2(t)z1(t)z2(t)
Stochastic
E[X1] = kE[Z1]− γrE[X1]
E[X2] = kpE[X1]− γpE[X2]
E[Z1] = µ− ηE[Z1]E[Z2]−ηV (Z1, Z2)
E[Z2] = E[X2]− ηE[Z1]E[Z2]−ηV (Z1, Z2)
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
Time
Pop
ulation
averages
[Molecules]
E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Robustness - Perfect adaptation
0 10 20 30 40 50 60 700
2
4
6
8
10
12
Time
Pop
ulation
averages
[Molecules]
E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]
(a) Perturbation of the controller gain k
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Time
Pop
ulation
averages
[Molecules]
E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]
(b) Perturbation of the translation rate kp
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
7
8
9
10
Time
Pop
ulation
averages
[Molecules]
E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]
(c) Perturbation of the mRNA degradation rate
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Time
Pop
ulation
averages
[Molecules]
E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]
(d) Perturbation of the protein degradation rate
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Concluding statements
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Concluding statements
What has been done• In-vivo (integral) control motif seems promising• Population control• Perfect adaptation
What needs to be done• Implementation• Extensions: bimolecular networks, different inputs, multiple inputs/outputs,
different control motifs→ biomolecular control theory
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Concluding statements
What has been done• In-vivo (integral) control motif seems promising• Population control• Perfect adaptation
What needs to be done• Implementation• Extensions: bimolecular networks, different inputs, multiple inputs/outputs,
different control motifs→ biomolecular control theory
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Introduction In-vivo control - Theory In-vivo control - Example Conclusion
Thank you for your attention
Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 14/14
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Computational results
TheoremThe following statements are equivalent:
(a) The matrix A is Hurwitz and the triplet (A, e1, eT` ) is output-controllable.
(b) There exist v ∈ Rd>0 and w ∈ Rd≥0 with wT e1 > 0, wT e` > 0, such that
vTA < 0 and wTA+ eT` = 0.
Comments• Linear program• Can be robustified→ if A ∈ [A−, A+], then vT+A
+ < 0 and wT−A− + eT` = 0.
• Can be made structural→ A ∈ {, 0,⊕}d×d
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Computational results
TheoremThe following statements are equivalent:
(a) The matrix A is Hurwitz and the triplet (A, e1, eT` ) is output-controllable.
(b) There exist v ∈ Rd>0 and w ∈ Rd≥0 with wT e1 > 0, wT e` > 0, such that
vTA < 0 and wTA+ eT` = 0.
Comments• Linear program• Can be robustified→ if A ∈ [A−, A+], then vT+A
+ < 0 and wT−A− + eT` = 0.
• Can be made structural→ A ∈ {, 0,⊕}d×d
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Implementation
Bacterial DNA Plasmids
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