Sparse Jurdjevic-Quinn stabilization Francesco Rossi - Universit - - PowerPoint PPT Presentation

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Sparse Jurdjevic-Quinn stabilization

Francesco Rossi - Université d’Aix-Marseille, France http://www.LSIS.org/frossi/ Collaboration with M. Caponigro, B. Piccoli, E. Trélat Hybrid Dynamical Systems: Optimization, Stability and Applications Università di Trento - January 10th, 2017

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Outline

1 Motivation: crowd modeling and crowd control 2 Sparse Jurdjevic-Quinn stabilization in finite dimension 3 Sparse Jurdjevic-Quinn stabilization for mean-field equations 2 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Outline

1 Motivation: crowd modeling and crowd control 2 Sparse Jurdjevic-Quinn stabilization in finite dimension 3 Sparse Jurdjevic-Quinn stabilization for mean-field equations 2 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Outline

1 Motivation: crowd modeling and crowd control 2 Sparse Jurdjevic-Quinn stabilization in finite dimension 3 Sparse Jurdjevic-Quinn stabilization for mean-field equations 2 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of crowds

analysis

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of crowds

input analysis

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of crowds

input analysis

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of crowds

input

• utput

analysis

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of crowds

input

• utput

analysis

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of crowds

input

• utput

control analysis

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Crowd modeling

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Crowd modeling

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Crowd modeling

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Crowd modeling

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Crowd modeling

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

For crowd applications, one needs

Sparse control

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

For crowd applications, one needs

Sparse control

Control needs to have bounded intensity, but also:

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

For crowd applications, one needs

Sparse control

Control needs to have bounded intensity, but also:

1 either act on a small set of agents (e.g. leaders); 5 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

For crowd applications, one needs

Sparse control

Control needs to have bounded intensity, but also:

1 either act on a small set of agents (e.g. leaders); 2 or act on a small set of the configuration space

(e.g. localized policy).

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

For crowd applications, one needs

Sparse control

Control needs to have bounded intensity, but also:

1 either act on a small set of agents (e.g. leaders);

good definition for finite-dimensional models.

2 or act on a small set of the configuration space

(e.g. localized policy).

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

For crowd applications, one needs

Sparse control

Control needs to have bounded intensity, but also:

1 either act on a small set of agents (e.g. leaders);

good definition for finite-dimensional models.

2 or act on a small set of the configuration space

(e.g. localized policy). good definition for mean-field models.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Section 1 Sparse Jurdjevic-Quinn stabilization in finite dimension

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

The classical Jurdjevic-Quinn stabilization method

Theorem (Jurdjevic-Quinn) Consider the system in Rd ˙ x(t) = f(x(t)) + m

i=1 ui(t)gi(x(t)).

(S) Assume that there exists a function V : Rd → R tha satisfies V is proper, i.e., V −1((−∞, ℓ]) is compact for every ℓ ∈ R; LfV (x) ≤ 0 for every x ∈ Rd. Then the smooth feedback defined by u(x) = −(Lg1V (x), Lg2V (x), . . . , LgmV (x)) asymptotically stabilizes (S) to the largest invariant subset of {x ∈ Rd | LfV (x) = Lk

fLgiV (x) = 0, for i = 1, . . . , m, k ≥ 0}.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

The classical Jurdjevic-Quinn stabilization method

The proof is based on LaSalle principle (not available in infinite dimension). The stabilizing control u(x) = −(Lg1V (x), Lg2V (x), . . . , LgmV (x)) is feedback, and many components are active.

Our goal is “component-wise sparsity”: at most one non-zero component at each time.

The target {x ∈ Rd | LfV (x) = Lk

fLgiV (x) = 0, for i = 1, . . . , m, k ≥ 0}

contains derivatives Lf of any order.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

The classical Jurdjevic-Quinn stabilization method

The proof is based on LaSalle principle (not available in infinite dimension). The stabilizing control u(x) = −(Lg1V (x), Lg2V (x), . . . , LgmV (x)) is feedback, and many components are active.

Our goal is “component-wise sparsity”: at most one non-zero component at each time.

The target {x ∈ Rd | LfV (x) = Lk

fLgiV (x) = 0, for i = 1, . . . , m, k ≥ 0}

contains derivatives Lf of any order.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

The classical Jurdjevic-Quinn stabilization method

The proof is based on LaSalle principle (not available in infinite dimension). The stabilizing control u(x) = −(Lg1V (x), Lg2V (x), . . . , LgmV (x)) is feedback, and many components are active.

Our goal is “component-wise sparsity”: at most one non-zero component at each time.

The target {x ∈ Rd | LfV (x) = Lk

fLgiV (x) = 0, for i = 1, . . . , m, k ≥ 0}

contains derivatives Lf of any order.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

The classical Jurdjevic-Quinn stabilization method

The proof is based on LaSalle principle (not available in infinite dimension). The stabilizing control u(x) = −(Lg1V (x), Lg2V (x), . . . , LgmV (x)) is feedback, and many components are active.

Our goal is “component-wise sparsity”: at most one non-zero component at each time.

The target {x ∈ Rd | LfV (x) = Lk

fLgiV (x) = 0, for i = 1, . . . , m, k ≥ 0}

contains derivatives Lf of any order.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

The simplest example on R2

˙ x ˙ y

• = u1g1 + u2g2 = u1

1

• + u2

1

• Exercise

One can stabilize the system to (0, 0) by choosing the functional V = x2 + y2. f = 0, hence LfV = 0; Lg1V = Lg2V = 0 in (0, 0) only; Jurdjevic-Quinn gives u(x, y) = (−2x, −2y), What happens when adding the component-wise sparsity constraint?

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

The simplest example on R2

˙ x ˙ y

• = u1g1 + u2g2 = u1

1

• + u2

1

• Exercise

One can stabilize the system to (0, 0) by choosing the functional V = x2 + y2. f = 0, hence LfV = 0; Lg1V = Lg2V = 0 in (0, 0) only; Jurdjevic-Quinn gives u(x, y) = (−2x, −2y), What happens when adding the component-wise sparsity constraint?

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

The simplest example on R2

˙ x ˙ y

• = u1g1 + u2g2 = u1

1

• + u2

1

• Exercise

One can stabilize the system to (0, 0) by choosing the functional V = x2 + y2. f = 0, hence LfV = 0; Lg1V = Lg2V = 0 in (0, 0) only; Jurdjevic-Quinn gives u(x, y) = (−2x, −2y), What happens when adding the component-wise sparsity constraint?

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Hybrid systems to prevent chattering

The component-wise sparsity constraint itself produces switching rules, that naturally introduce chattering phenomena. We present three possible solutions: Strategies 1 and 2 are based on time sampling; Strategy 3 is based on hysteresis.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Hybrid systems to prevent chattering

The component-wise sparsity constraint itself produces switching rules, that naturally introduce chattering phenomena. We present three possible solutions: Strategies 1 and 2 are based on time sampling; Strategy 3 is based on hysteresis.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations 11 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations 11 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1: Sparse time-varying feedback

Fix the sampling time τ > 0. For the initial state x0 ∈ Rn, consider the trajectory x(t) of (S) with the time-varying feedback control u(t, x) defined as follows:

• n each time interval [(km + i)τ, (km + i + 1)τ),

apply the feedback control u(t, x) = −LgiV (x(t))ei. Remark This is a kind of averaged control.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1: Sparse time-varying feedback

Fix the sampling time τ > 0. For the initial state x0 ∈ Rn, consider the trajectory x(t) of (S) with the time-varying feedback control u(t, x) defined as follows:

• n each time interval [(km + i)τ, (km + i + 1)τ),

apply the feedback control u(t, x) = −LgiV (x(t))ei. Remark This is a kind of averaged control.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1: Sparse time-varying feedback

Theorem (Caponigro-Piccoli-Rossi-Trélat, submitted) Let Z = {x ∈ Rn | LfV (x) = LgiV (x) = 0, for every i = 1, . . . , m}, and Ω be the largest subset of Z invariant for ˙ x = f(x). If Ω is locally attractive, then, for each initial state x0 ∈ Rn, there exist τ1 > 0 such that, for every τ ∈ (0, τ1), Strategy 1 with sampling time τ stabilizes the control system (S) to Ω. Remark Local attractiveness is necessary, see counterexamples.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1: Sparse time-varying feedback

Theorem (Caponigro-Piccoli-Rossi-Trélat, submitted) Let Z = {x ∈ Rn | LfV (x) = LgiV (x) = 0, for every i = 1, . . . , m}, and Ω be the largest subset of Z invariant for ˙ x = f(x). If Ω is locally attractive, then, for each initial state x0 ∈ Rn, there exist τ1 > 0 such that, for every τ ∈ (0, τ1), Strategy 1 with sampling time τ stabilizes the control system (S) to Ω. Remark Local attractiveness is necessary, see counterexamples.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1: Sparse time-varying feedback

τ = 1

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1: Sparse time-varying feedback

τ = 0.1

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1: Sparse time-varying feedback

τ = 0.01

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1: Sparse time-varying feedback

Strategy 1 is far from being “optimal” in some sense;

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1: Sparse time-varying feedback

Strategy 1 is far from being “optimal” in some sense; It is useful on long-range, not locally.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations 15 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations 15 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2: Sampled sparse feedback

Fix a sampling time τ > 0. In x0, choose the smallest integer i ∈ {1, . . . , m} such that |LgiV (x0)| ≥ |LgjV (x0)|, ∀j = i. Then consider the sampling solution associated with u and the sampling time τ until x1 = x(τ). Compute the new control in x1, and interatively for each xn. Remark The control is chosen as a “steepest descent” for V , then it is kept for the whole time interval [0, τ].

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2: Sampled sparse feedback

Fix a sampling time τ > 0. In x0, choose the smallest integer i ∈ {1, . . . , m} such that |LgiV (x0)| ≥ |LgjV (x0)|, ∀j = i. Then consider the sampling solution associated with u and the sampling time τ until x1 = x(τ). Compute the new control in x1, and interatively for each xn. Remark The control is chosen as a “steepest descent” for V , then it is kept for the whole time interval [0, τ].

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2: Sampled sparse feedback

Theorem (CPRT) If {x ∈ Rn | LfV (x) = LgiV (x) = 0, for every i = 1, . . . , m} = {¯ x} for some ¯ x ∈ Rn, then, for every initial state x0 ∈ Rn, there exists τ2 > 0 such that, for every τ ∈ (0, τ2), Strategy 2 with sampling time τ practically asymptotically stabilizes (S) to ¯ x. Remark The functional V (t) is not strictly decreasing, one can have problems for Z not reduced to a point.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2: Sampled sparse feedback

Theorem (CPRT) If {x ∈ Rn | LfV (x) = LgiV (x) = 0, for every i = 1, . . . , m} = {¯ x} for some ¯ x ∈ Rn, then, for every initial state x0 ∈ Rn, there exists τ2 > 0 such that, for every τ ∈ (0, τ2), Strategy 2 with sampling time τ practically asymptotically stabilizes (S) to ¯ x. Remark The functional V (t) is not strictly decreasing, one can have problems for Z not reduced to a point.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2: Sparse time-varying feedback

τ = 1

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2: Sparse time-varying feedback

τ = 0.5

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2: Sparse time-varying feedback

τ = 0.01

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2: Sparse time-varying feedback

Strategy 2 is efficient along axes. It is not efficient around the switching set.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

Fix ε ∈ (0, 1) and apply the following algorithm: Initialize: n = 0 and t0 = 0. While tn < +∞ apply Step n: At time tn choose i = 1, . . . , m the smallest integer such that |LgiV (x(tn))| ≥ |LgjV (x(tn))|, for every j = i.

If |LgiV (x(tn))| ≥ 2t−1

n , keep the control u = −LgiV (x) until

|LgiV (x(t))| ≤ t−1

• r

|LgiV x(t))| ≤ (1 − ε)|LgjV (x(t))| for some j = 1, . . . , m. If |LgiV (x(tn))| < 2t−1

n , keep the control u = 0 until

|LgjV (x(t))| ≥ 4t−1, for some j = 1, . . . , m.

Remark The control is chosen as a “steepest descent” for V , then continuously evaluated along time.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

Fix ε ∈ (0, 1) and apply the following algorithm: Initialize: n = 0 and t0 = 0. While tn < +∞ apply Step n: At time tn choose i = 1, . . . , m the smallest integer such that |LgiV (x(tn))| ≥ |LgjV (x(tn))|, for every j = i.

If |LgiV (x(tn))| ≥ 2t−1

n , keep the control u = −LgiV (x) until

|LgiV (x(t))| ≤ t−1

• r

|LgiV x(t))| ≤ (1 − ε)|LgjV (x(t))| for some j = 1, . . . , m. If |LgiV (x(tn))| < 2t−1

n , keep the control u = 0 until

|LgjV (x(t))| ≥ 4t−1, for some j = 1, . . . , m.

Remark The control is chosen as a “steepest descent” for V , then continuously evaluated along time.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

Fix ε ∈ (0, 1) and apply the following algorithm: Initialize: n = 0 and t0 = 0. While tn < +∞ apply Step n: At time tn choose i = 1, . . . , m the smallest integer such that |LgiV (x(tn))| ≥ |LgjV (x(tn))|, for every j = i.

If |LgiV (x(tn))| ≥ 2t−1

n , keep the control u = −LgiV (x) until

|LgiV (x(t))| ≤ t−1

• r

|LgiV x(t))| ≤ (1 − ε)|LgjV (x(t))| for some j = 1, . . . , m. If |LgiV (x(tn))| < 2t−1

n , keep the control u = 0 until

|LgjV (x(t))| ≥ 4t−1, for some j = 1, . . . , m.

Remark The control is chosen as a “steepest descent” for V , then continuously evaluated along time.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

Fix ε ∈ (0, 1) and apply the following algorithm: Initialize: n = 0 and t0 = 0. While tn < +∞ apply Step n: At time tn choose i = 1, . . . , m the smallest integer such that |LgiV (x(tn))| ≥ |LgjV (x(tn))|, for every j = i.

If |LgiV (x(tn))| ≥ 2t−1

n , keep the control u = −LgiV (x) until

|LgiV (x(t))| ≤ t−1

• r

|LgiV x(t))| ≤ (1 − ε)|LgjV (x(t))| for some j = 1, . . . , m. If |LgiV (x(tn))| < 2t−1

n , keep the control u = 0 until

|LgjV (x(t))| ≥ 4t−1, for some j = 1, . . . , m.

Remark The control is chosen as a “steepest descent” for V , then continuously evaluated along time.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

Theorem (CPRT) For any ε ∈ (0, 1), Strategy 3 asymptotically stabilizes (S) to Ω, the largest invariant subset of {x ∈ Rn | LfV (x) = LgiV (x) = 0, for every i = 1, . . . , m}. Remark The strategy is more robust, since it works for any ε. One needs a continuous evaluation of the control.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

Theorem (CPRT) For any ε ∈ (0, 1), Strategy 3 asymptotically stabilizes (S) to Ω, the largest invariant subset of {x ∈ Rn | LfV (x) = LgiV (x) = 0, for every i = 1, . . . , m}. Remark The strategy is more robust, since it works for any ε. One needs a continuous evaluation of the control.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

Theorem (CPRT) For any ε ∈ (0, 1), Strategy 3 asymptotically stabilizes (S) to Ω, the largest invariant subset of {x ∈ Rn | LfV (x) = LgiV (x) = 0, for every i = 1, . . . , m}. Remark The strategy is more robust, since it works for any ε. One needs a continuous evaluation of the control.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

ε = 0.5

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

ε = 0.1

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

ε = 0.01

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3: Sparse feedback with hysteresis

Similarly to Strategy 2, Strategy 3 is efficient along axes. It is not efficient around the switching set.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Comparison between strategies

2 4 6 8 2 4 6 8

Orange: “Time to target=1” for Strategy 1 for τ → 0. Red: Strategy 2 for τ → 0 and Strategy 3 for ε → 0. Green: Non-sparse Jurdjevic-Quinn.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Comparison between strategies

2 4 6 8 2 4 6 8

Orange: “Time to target=1” for Strategy 1 for τ → 0. Red: Strategy 2 for τ → 0 and Strategy 3 for ε → 0. Green: Non-sparse Jurdjevic-Quinn.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Sparse control of the Hegselmann-Krause model

Each agent has an opinion xi(t), that evolves by interaction with his neighbours only: ˙ xi =

• |xj−xi|<1

(xj − xi)

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Sparse control of the Hegselmann-Krause model

Each agent has an opinion xi(t), that evolves by interaction with his neighbours only: ˙ xi =

• |xj−xi|<1

(xj − xi) + ui.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Sparse control of the Hegselmann-Krause model

Each agent has an opinion xi(t), that evolves by interaction with his neighbours only: ˙ xi =

• |xj−xi|<1

(xj − xi) + ui. We aim to enforce consensus by minimizing the variance V =

• i,j

(xi − xj)2.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Sparse control of the Hegselmann-Krause model

Each agent has an opinion xi(t), that evolves by interaction with his neighbours only: ˙ xi =

• |xj−xi|<1

(xj − xi) + ui. We aim to enforce consensus by minimizing the variance V =

• i,j

(xi − xj)2. LfV ≤ 0, thus we can apply Jurdjevic-Quinn.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Simulation with 50 randomized agents in [0, 10]

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1 for Hegselmann-Krause model

τ = 1

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1 for Hegselmann-Krause model

τ = 0.5

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 1 for Hegselmann-Krause model

τ = 0.1

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2 for Hegselmann-Krause model

τ = 1

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2 for Hegselmann-Krause model

τ = 0.5

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 2 for Hegselmann-Krause model

τ = 0.1

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3 for Hegselmann-Krause model

ε = 0.2

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3 for Hegselmann-Krause model

ε = 0.05

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Strategy 3 for Hegselmann-Krause model

ε = 0.01

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Section 2 Sparse Jurdjevic-Quinn stabilization for mean-field equations

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Mean-Field equations

Let us consider a large number N of interacting agents: Analysis of a very large-dimensional ODE is hard; Numerical simulations are hard; Control is hard. If agents are all identical, we can replace the positions of agents with the density of the crowd µ. What is the dynamics of µ?

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Mean-Field equations

Let us consider a large number N of interacting agents: Analysis of a very large-dimensional ODE is hard; Numerical simulations are hard; Control is hard. If agents are all identical, we can replace the positions of agents with the density of the crowd µ. What is the dynamics of µ?

29 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Mean-Field equations

Let us consider a large number N of interacting agents: Analysis of a very large-dimensional ODE is hard; Numerical simulations are hard; Control is hard. If agents are all identical, we can replace the positions of agents with the density of the crowd µ. What is the dynamics of µ?

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Proposition If f is regular, the mean-field limit of ˙ xi = 1 N

• j=i

f(xj − xi) (ODE) is ∂tµ + ∇ · (F[µ]µ) = 0. (PDE) where F[µ] = f ⋆ µ. Conditions of existence, uniqueness, continuous dependence...?

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Proposition If f is regular, the mean-field limit of ˙ xi = 1 N

• j=i

f(xj − xi) (ODE) is ∂tµ + ∇ · (F[µ]µ) = 0. (PDE) where F[µ] = f ⋆ µ. Conditions of existence, uniqueness, continuous dependence...?

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Proposition Assume that F is defined by F [µ] (x) := v(f ⋆ µ(x)) with v, f Lipschitz. Then there exists a unique solution of the Cauchy problem

• ∂tµ + ∇ · (F[µ] µ) = 0

µ|t=0 = µ0, Remark This is a particular case of a theory in Wasserstein spaces: Ambrosio-Gangbo, Colombo-Garavello-Mercier, Piccoli-Rossi... Proposition (Piccoli-Tosin 2011, Piccoli-Rossi 2013) Numerical schemes are available.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the transport equation

N interacting particles ˙ xi = 1

N

• j f(xi − xj)

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the transport equation

N interacting particles ˙ xi = 1

N

• j f(xi − xj)

Control of the density

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the transport equation

N interacting particles ˙ xi = 1

N

• j f(xi − xj)

mean-field − − − − − − → Evolution of the density ∂tµ + ∇ · (F[µ]µ) = 0 Control of the density

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the transport equation

N interacting particles ˙ xi = 1

N

• j f(xi − xj)

mean-field − − − − − − → Evolution of the density ∂tµ + ∇ · (F[µ]µ) = 0 ↓ control? Control of the density

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the transport equation

N interacting particles ˙ xi = 1

N

• j f(xi − xj)

mean-field − − − − − − → Evolution of the density ∂tµ + ∇ · (F[µ]µ) = 0 control ↓ ↓ control? Controlled ODE ˙ xi = 1

N

• j f(xi − xj)+ui(t)

Control of the density

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the transport equation

N interacting particles ˙ xi = 1

N

• j f(xi − xj)

mean-field − − − − − − → Evolution of the density ∂tµ + ∇ · (F[µ]µ) = 0 control ↓ ↓ control? Controlled ODE ˙ xi = 1

N

• j f(xi − xj)+ui(t)

mean-field? − − − − − − → Control of the density

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the transport equation

How to control the transport equation? How to define controls? Remark Controls on each agent

• ˙

xi =

j f(xj − xi)+ui(t)

make no sense in the mean-field limit. Definition The control is a localized vector field χωu. It acts as follows ∂tµ + ∇ · [(F [µ] +χωu)µ] = 0. We need χωu Lipschitz.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the transport equation

How to control the transport equation? How to define controls? Remark Controls on each agent

• ˙

xi =

j f(xj − xi)+ui(t)

make no sense in the mean-field limit. Definition The control is a localized vector field χωu. It acts as follows ∂tµ + ∇ · [(F [µ] +χωu)µ] = 0. We need χωu Lipschitz.

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Definition (Sparsity constraint) Fix a small area c > 0 of the configuration space and impose the control set ω(t) satisfies |ω(t)| ≤ c; the control function u(t) is defined on ω(t) with u(t)∞ ≤ 1 We aim to generalize Strategy 3 to this setting. Remark One control only, for more controls adapt finite-dimensional. The hardest part is to understand how to generalize objets for the Jurdjevic-Quinn method: proper Lyapunov function, Lie derivatives, steepest descent...

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Definition (Sparsity constraint) Fix a small area c > 0 of the configuration space and impose the control set ω(t) satisfies |ω(t)| ≤ c; the control function u(t) is defined on ω(t) with u(t)∞ ≤ 1 We aim to generalize Strategy 3 to this setting. Remark One control only, for more controls adapt finite-dimensional. The hardest part is to understand how to generalize objets for the Jurdjevic-Quinn method: proper Lyapunov function, Lie derivatives, steepest descent...

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Definition (Sparsity constraint) Fix a small area c > 0 of the configuration space and impose the control set ω(t) satisfies |ω(t)| ≤ c; the control function u(t) is defined on ω(t) with u(t)∞ ≤ 1 We aim to generalize Strategy 3 to this setting. Remark One control only, for more controls adapt finite-dimensional. The hardest part is to understand how to generalize objets for the Jurdjevic-Quinn method: proper Lyapunov function, Lie derivatives, steepest descent...

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

The idea is to have Lipschitz-mollified characteristic functions χη

[a,b]

b + η b a a − η

Ωt =

• (a, b, η) | |ω(a, b, η)| ≤ c and η ≥ t−1

. At time tn, choose one of the maximizers (a∗, b∗, η∗) in Ωtn of stn(a, b, η) := |Lχη

[a,b]gV [µ(tn)]| =

• lim

τ→0

V [eτχη

[a,b]gµ(tn)] − V [µ(tn)]

τ

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

If either stn(a∗, b∗, η∗) < t−1

n

• r Ωtn is empty, then choose the

zero control. Let the measure evolve up to the existence of (¯ a,¯ b, ¯ η) ∈ Ω′

t

such that st(¯ a,¯ b, ¯ η) ≥ 2t−1. If stn(a∗, b∗, η∗) ≥ t−1

n , then choose the control

u(t, ·) = −χη∗

[a∗,b∗] sign(Lχη∗

[a∗,b∗]g[µ(t)]V [µ(t)]).

Let the measure evolve up to one of the following conditions:

either st(a∗, b∗, η∗) ≤ t−1

2 ;

• r there exists (¯

a,¯ b, ¯ η) ∈ Ω′

t such that

st(a∗, b∗, η∗) ≤ (1 − ε)st(¯ a,¯ b, ¯ η).

Theorem (CPRT, submitted) For each ε ∈ (0, 1), this strategy drives the solution to (PDE) to Z = {µ ∈ Pc(Rn) | LfV [µ] = LugV [µ] = 0 ∀u ∈ Lip(Rn, R)}

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the mean-field Hegselmann-Krause model

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the mean-field Hegselmann-Krause model

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the mean-field Hegselmann-Krause model

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the mean-field Hegselmann-Krause model

ε = 0.9

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the mean-field Hegselmann-Krause model

ε = 0.5

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Control of the mean-field Hegselmann-Krause model

ε = 0.2

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Conclusions

In this talk, we showed: the interest of sparse control, in particular for control of crowd models; the use of hybrid methods (sampling and hysteresis) to solve stabilization problems with sparse constraint; the generalization of such techniques to mean-field models (for large crowds). Warmest thanks for your attention!

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Conclusions

In this talk, we showed: the interest of sparse control, in particular for control of crowd models; the use of hybrid methods (sampling and hysteresis) to solve stabilization problems with sparse constraint; the generalization of such techniques to mean-field models (for large crowds). Warmest thanks for your attention!

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Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Conclusions

In this talk, we showed: the interest of sparse control, in particular for control of crowd models; the use of hybrid methods (sampling and hysteresis) to solve stabilization problems with sparse constraint; the generalization of such techniques to mean-field models (for large crowds). Warmest thanks for your attention!

40 / 40

Sparse Jurdjevic-Quinn stabilization in finite dimension Sparse Jurdjevic-Quinn stabilization for mean-field equations

Conclusions

In this talk, we showed: the interest of sparse control, in particular for control of crowd models; the use of hybrid methods (sampling and hysteresis) to solve stabilization problems with sparse constraint; the generalization of such techniques to mean-field models (for large crowds). Warmest thanks for your attention!

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