Solvability of Matrix-Exponential Equations Jol Ouaknine, Amaury - PowerPoint PPT Presentation

Solvability of Matrix-Exponential Equations Joël Ouaknine, Amaury Pouly, João Sousa-Pinto, James Worrell University of Oxford

Example: 2D robot State: � u = ( x θ , y θ , x , y ) Discretized actions: ( x , y ) ◮ rotate arm by ψ ◮ change arm length by δ ℓ ( x θ , y θ ) → Linear transformations: θ u ← Au Rotate arm by ψ : Change arm length by δ : � � � � � � cos ψ − sin ψ x x ← y sin ψ cos ψ y � � � � � � x x x θ ← + δ y y y θ � � � � � � x θ cos ψ − sin ψ x θ ← sin ψ cos ψ y θ y θ

Linear dynamical systems Discrete case Continuous case x ′ ( t ) = Ax ( t ) x ( n + 1) = Ax ( n ) ◮ biology, ◮ biology, ◮ software verification, ◮ physics, ◮ probabilistic model ◮ probabilistic model checking, checking, ◮ combinatorics, ◮ electrical circuits, ◮ .... ◮ .... Typical questions ◮ reachability ◮ safety ◮ controllability

Related work in the discrete case Input: A , C ∈ Q d × d matrices Output: ∃ n ∈ N such that A n = C ? Example: ∃ n ∈ N such that � n � � � 1 1 1 100 = ? 0 1 0 1

Related work in the discrete case Input: A , C ∈ Q d × d matrices Output: ∃ n ∈ N such that A n = C ? � Decidable (PTIME) Example: ∃ n ∈ N such that � n � � � 1 1 1 100 = ? 0 1 0 1

Related work in the discrete case Input: A , C ∈ Q d × d matrices Output: ∃ n ∈ N such that A n = C ? � Decidable (PTIME) Input: A , B , C ∈ Q d × d matrices Output: ∃ n , m ∈ N such that A n B m = C ? Example: ∃ n , m ∈ N such that � n � 1 � m � 1 � � 2 3 1 60 2 2 = ? 0 1 0 1 0 1

Related work in the discrete case Input: A , C ∈ Q d × d matrices Output: ∃ n ∈ N such that A n = C ? � Decidable (PTIME) Input: A , B , C ∈ Q d × d matrices Output: ∃ n , m ∈ N such that A n B m = C ? � Decidable Example: ∃ n , m ∈ N such that � n � 1 � m � 1 � � 2 3 1 60 2 2 = ? 0 1 0 1 0 1

Related work in the discrete case Input: A , C ∈ Q d × d matrices Output: ∃ n ∈ N such that A n = C ? � Decidable (PTIME) Input: A , B , C ∈ Q d × d matrices Output: ∃ n , m ∈ N such that A n B m = C ? � Decidable Input: A 1 , . . . , A k , C ∈ Q d × d matrices i =1 A n i Output: ∃ n 1 , . . . , n k ∈ N such that � k i = C ? Example: ∃ n , m , p ∈ N such that � n � 1 � m � � p � 1 � � 2 3 2 5 81 260 2 2 = ? 0 1 0 1 0 1 0 1

Related work in the discrete case Input: A , C ∈ Q d × d matrices Output: ∃ n ∈ N such that A n = C ? � Decidable (PTIME) Input: A , B , C ∈ Q d × d matrices Output: ∃ n , m ∈ N such that A n B m = C ? � Decidable Input: A 1 , . . . , A k , C ∈ Q d × d matrices i =1 A n i Output: ∃ n 1 , . . . , n k ∈ N such that � k i = C ? � Decidable if A i commute × Undecidable in general Example: ∃ n , m , p ∈ N such that � n � 1 � m � � p � 1 � � 2 3 2 5 81 260 2 2 = ? 0 1 0 1 0 1 0 1

Related work in the discrete case Input: A , C ∈ Q d × d matrices Output: ∃ n ∈ N such that A n = C ? � Decidable (PTIME) Input: A , B , C ∈ Q d × d matrices Output: ∃ n , m ∈ N such that A n B m = C ? � Decidable Input: A 1 , . . . , A k , C ∈ Q d × d matrices i =1 A n i Output: ∃ n 1 , . . . , n k ∈ N such that � k i = C ? � Decidable if A i commute × Undecidable in general Input: A 1 , . . . , A k , C ∈ Q d × d matrices Output: C ∈ � semi-group generated by A 1 , . . . , A k � ? Semi-group: � A 1 , . . . , A k � = all finite products of A 1 , . . . , A k Examples: A 8 3 A 2 A 3 1 A 42 A 1 A 3 A 2 A 1 A 2 A 1 A 2 3

Related work in the discrete case Input: A , C ∈ Q d × d matrices Output: ∃ n ∈ N such that A n = C ? � Decidable (PTIME) Input: A , B , C ∈ Q d × d matrices Output: ∃ n , m ∈ N such that A n B m = C ? � Decidable Input: A 1 , . . . , A k , C ∈ Q d × d matrices i =1 A n i Output: ∃ n 1 , . . . , n k ∈ N such that � k i = C ? � Decidable if A i commute × Undecidable in general Input: A 1 , . . . , A k , C ∈ Q d × d matrices Output: C ∈ � semi-group generated by A 1 , . . . , A k � ? � Decidable if A i commute × Undecidable in general Semi-group: � A 1 , . . . , A k � = all finite products of A 1 , . . . , A k Examples: A 8 3 A 2 A 3 1 A 42 A 1 A 3 A 2 A 1 A 2 A 1 A 2 3

Recap on linear differential equations Let x : R + → R n function, A ∈ Q n × n matrix x 1 ( t ) a 11 · · · a 1 n     . . . ... . . . x ( t ) = A =     . . .     x n ( t ) · · · a n 1 a nn Linear differential equation: x ′ ( t ) = Ax ( t ) x (0) = x 0

Recap on linear differential equations Let x : R + → R n function, A ∈ Q n × n matrix x 1 ( t ) a 11 · · · a 1 n     . . . ... . . . x ( t ) = A =     . . .     x n ( t ) · · · a n 1 a nn Linear differential equation: x ′ ( t ) = Ax ( t ) x (0) = x 0 Examples: � x ′ 1 ( t )= x 2 ( t ) x ′ ( t ) = 7 x ( t ) x ′ 2 ( t )= − x 1 ( t ) ❀ x ( t ) = e 7 t � x 1 ( t )= sin( t ) ❀ x 2 ( t )= cos( t )

Recap on linear differential equations Let x : R + → R n function, A ∈ Q n × n matrix x 1 ( t ) a 11 · · · a 1 n     . . . ... . . . x ( t ) = A =     . . .     x n ( t ) · · · a n 1 a nn Linear differential equation: x ′ ( t ) = Ax ( t ) x (0) = x 0 Examples: � ′ � � � � � � x ′ 1 ( t )= x 2 ( t ) 0 1 x 1 x 1 x ′ ( t ) = 7 x ( t ) ⇔ = x ′ 2 ( t )= − x 1 ( t ) x 2 − 1 0 x 2 ❀ x ( t ) = e 7 t � x 1 ( t )= sin( t ) ❀ x 2 ( t )= cos( t )

Recap on linear differential equations Let x : R + → R n function, A ∈ Q n × n matrix x 1 ( t ) a 11 · · · a 1 n     . . . ... . . . x ( t ) = A =     . . .     x n ( t ) · · · a n 1 a nn Linear differential equation: x ′ ( t ) = Ax ( t ) x (0) = x 0 General solution form: x ( t ) = exp( At ) x 0 ∞ M n � where exp( M ) = n ! n =0

Related work in the continuous case Input: A , C ∈ Q d × d matrices Output: ∃ t ∈ R such that e At = C ? Example: ∃ t ∈ R such that �� � � � � 1 1 1 100 exp = ? t 0 1 0 1

Related work in the continuous case Input: A , C ∈ Q d × d matrices Output: ∃ t ∈ R such that e At = C ? � Decidable (PTIME) Example: ∃ t ∈ R such that �� � � � � 1 1 1 100 exp = ? t 0 1 0 1

Related work in the continuous case Input: A , C ∈ Q d × d matrices Output: ∃ t ∈ R such that e At = C ? � Decidable (PTIME) Input: A , B , C ∈ Q d × d matrices Output: ∃ t , u ∈ N such that e At e Bu = C ? Example: ∃ t , u ∈ R such that �� 1 �� � � 1 � � � � 2 3 1 60 2 2 exp exp = ? t u 0 1 0 1 0 1

Related work in the continuous case Input: A , C ∈ Q d × d matrices Output: ∃ t ∈ R such that e At = C ? � Decidable (PTIME) Input: A , B , C ∈ Q d × d matrices Output: ∃ t , u ∈ N such that e At e Bu = C ? × Unknown Example: ∃ t , u ∈ R such that �� � � �� 1 1 � � � � 2 3 1 60 2 2 exp exp = ? t u 0 1 0 1 0 1

Hybrid/Cyber-physical systems ◮ physics: continuous dynamics ◮ electronics: discrete states state continuous dynamics guard φ ( x ) x ′ = F 1 ( x ) x ′ = F 2 ( x ) x ← R ( x ) discrete update

Hybrid/Cyber-physical systems ◮ physics: continuous dynamics ◮ electronics: discrete states state continuous dynamics guard φ ( x ) x ′ = F 1 ( x ) x ′ = F 2 ( x ) x ← R ( x ) discrete update Some classes: Typical questions ◮ F i ( x ) = 1: timed automata ◮ reachability ◮ F i ( x ) = c i : rectangular hybrid automata ◮ safety ◮ F i ( x ) = A i x : linear hybrid automata ◮ controllability

Switching system x ′ = A 1 x x ′ = A 2 x Restricted hybrid system: ◮ linear dynamics ◮ no guards (nondeterministic) x ′ = A 4 x x ′ = A 3 x ◮ no discrete updates switch x 1 ( t ) t x ′ = A 1 x x ′ = A 2 x x ′ = A 3 x x ′ = A 4 x t 1 t 2 t 3 t 4

Switching system x ′ = A 1 x x ′ = A 2 x Restricted hybrid system: ◮ linear dynamics ◮ no guards (nondeterministic) x ′ = A 4 x x ′ = A 3 x ◮ no discrete updates switch x 1 ( t ) t x ′ = A 1 x x ′ = A 2 x x ′ = A 3 x x ′ = A 4 x t 1 t 2 t 3 t 4 Dynamics: e A 4 t 4 e A 3 t 3 e A 2 t 2 e A 1 t 1

Switching system x ′ = A 1 x x ′ = A 2 x Restricted hybrid system: ◮ linear dynamics ◮ no guards (nondeterministic) x ′ = A 4 x x ′ = A 3 x ◮ no discrete updates switch x 1 ( t ) t x ′ = A 1 x x ′ = A 2 x x ′ = A 3 x x ′ = A 4 x t 1 t 2 t 3 t 4 Problem: e A 4 t 4 e A 3 t 3 e A 2 t 2 e A 1 t 1 = C ? What we control: t 1 , t 2 , t 3 , t 4 ∈ R +

Switching system x ′ = A 1 x x ′ = A 2 x What about a loop ? x ′ = A 4 x x ′ = A 3 x

Switching system x ′ = A 1 x x ′ = A 2 x What about a loop ? x ′ = A 4 x x ′ = A 3 x x 1 ( t ) A 1 A 2 A 3 A 4 A 1 A 2 A 3 A 4 t t 1 t 2 t 3 t 4 t ′ t ′ t ′ t ′ 1 2 3 4 Dynamics: e A 4 t ′ 4 e A 3 t ′ 3 e A 2 t ′ 2 e A 1 t ′ 1 e A 4 t 4 e A 3 t 3 e A 2 t 2 e A 1 t 1