On the Decidability of Reachability in Linear Time-Invariant Systems - PowerPoint PPT Presentation

On the Decidability of Reachability in Linear Time-Invariant Systems Nathanaël Fijalkow, Joël Ouaknine, Amaury Pouly, João Sousa-Pinto, James Worrell Université de Paris, IRIF, CNRS 26 november 2019 1 / 12

Example : mass-spring-damper system State : X = z ∈ R Equation of motion : b k mz ′′ = − kz − bz ′ + mg + u m u ( t ) Model with external input u ( t ) 2 / 12

Example : mass-spring-damper system State : X = z ∈ R Equation of motion : z b k mz ′′ = − kz − bz ′ + mg + u m u ( t ) Model with external input u ( t ) 2 / 12

Example : mass-spring-damper system State : X = z ∈ R Equation of motion : z b k mz ′′ = − kz − bz ′ + mg + u → Affine but not first order m u ( t ) Model with external input u ( t ) 2 / 12

Example : mass-spring-damper system State : X = z ∈ R Equation of motion : z b k mz ′′ = − kz − bz ′ + mg + u → Affine but not first order m u ( t ) State : X = ( z , z ′ , 1 ) ∈ R 3 Model with external input u ( t ) Equation of motion : ′     z z ′ m z ′ + g + 1 − k m z − b z ′ = m u     1 0 2 / 12

Example : mass-spring-damper system State : X = z ∈ R Equation of motion : z b k mz ′′ = − kz − bz ′ + mg + u → Affine but not first order m u ( t ) State : X = ( z , z ′ , 1 ) ∈ R 3 Model with external input u ( t ) Equation of motion : → Linear time invariant system ′     z z ′ X ′ = AX + Bu m z ′ + g + 1 − k m z − b z ′ = m u     1 0 with some constraints on u . 2 / 12

Linear dynamical systems Discrete case Continuous case x ( n + 1 ) = Ax ( n ) x ′ ( t ) = Ax ( t ) ◮ biology, ◮ biology, ◮ software verification, ◮ physics, ◮ probabilistic model checking, ◮ probabilistic model checking, ◮ combinatorics, ◮ electrical circuits, ◮ .... ◮ .... Typical questions ◮ reachability ◮ safety 3 / 12

Linear dynamical systems Discrete case Continuous case x ( n + 1 ) = Ax ( n ) + Bu ( n ) x ′ ( t ) = Ax ( t ) + Bu ( t ) ◮ biology, ◮ biology, ◮ software verification, ◮ physics, ◮ probabilistic model checking, ◮ probabilistic model checking, ◮ combinatorics, ◮ electrical circuits, ◮ .... ◮ .... Typical questions ◮ reachability ◮ safety ◮ controllability 3 / 12

Linear dynamical systems Discrete case Continuous case x ( n + 1 ) = Ax ( n ) + Bu ( n ) x ′ ( t ) = Ax ( t ) + Bu ( t ) ◮ biology, ◮ biology, ◮ software verification, ◮ physics, ◮ probabilistic model checking, ◮ probabilistic model checking, ◮ combinatorics, ◮ electrical circuits, ◮ .... ◮ .... Typical questions ◮ reachability ◮ optimal control ◮ safety ◮ feedback control ◮ controllability ◮ ... 3 / 12

The problem LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . s t 4 / 12

The problem LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . x 0 = s t 4 / 12

The problem LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . Ax 0 x 0 = s t 4 / 12

The problem LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . x 1 = Ax 0 + u 0 u 0 Ax 0 x 0 = s t 4 / 12

The problem LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . x 1 = Ax 0 + u 0 u 0 Ax 0 x 0 = s t Ax 1 4 / 12

The problem LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . x 1 = Ax 0 + u 0 u 0 Ax 0 x 2 = Ax 1 + u 1 x 0 = s t u 1 Ax 1 4 / 12

The problem LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . x 1 = Ax 0 + u 0 Ax 2 u 0 Ax 0 x 2 = Ax 1 + u 1 x 0 = s t u 1 Ax 1 4 / 12

The problem LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . x 1 = Ax 0 + u 0 Ax 2 u 0 Ax 0 u 2 x 2 = Ax 1 + u 1 x 0 = s x 3 = t u 1 Ax 1 4 / 12

Existing work LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . 5 / 12

Existing work LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . Theorem (Lipton and Kannan, 1986) LTI-REACHABILITY is decidable if U is an affine subspace of R d . 5 / 12

Existing work LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . Theorem (Lipton and Kannan, 1986) LTI-REACHABILITY is decidable if U is an affine subspace of R d . Almost no exact results for other classes of U 5 / 12

Existing work LTI-REACHABILITY ◮ a source s ∈ Q d , ◮ a target t ∈ Q d , ◮ a transition matrix A ∈ Q d × d , ◮ a set of controls U ⊆ R d , decide if ∃ T ∈ N , u 0 , . . . , u T − 1 ∈ U such that x T = t where x 0 = s , x n + 1 = Ax n + u n . Theorem (Lipton and Kannan, 1986) LTI-REACHABILITY is decidable if U is an affine subspace of R d . Almost no exact results for other classes of U in particular when U is bounded (which is the most natural case). 5 / 12

Our results : hardness Study the impact of the control set on the hardness of reachability 6 / 12

Our results : hardness Study the impact of the control set on the hardness of reachability Theorem LTI-REACHABILITY is ◮ undecidable if U is a finite union of affine subspaces. 6 / 12

Our results : hardness Study the impact of the control set on the hardness of reachability Theorem LTI-REACHABILITY is ◮ undecidable if U is a finite union of affine subspaces. ◮ Skolem-hard if U = { 0 } ∪ V where V is an affine subspace Given s ∈ Q d and A ∈ Q d × d : ◮ Skolem problem : decide if ∃ T ∈ N such that ( A T s ) 1 = 0, 6 / 12

Our results : hardness Study the impact of the control set on the hardness of reachability Theorem LTI-REACHABILITY is ◮ undecidable if U is a finite union of affine subspaces. ◮ Skolem-hard if U = { 0 } ∪ V where V is an affine subspace ◮ Positivity-hard if U is a convex polytope Given s ∈ Q d and A ∈ Q d × d : ◮ Skolem problem : decide if ∃ T ∈ N such that ( A T s ) 1 = 0, ◮ Positivity problem : decide if ( A T s ) 1 � 0 for all T ∈ N . 6 / 12

Our results : hardness Study the impact of the control set on the hardness of reachability Theorem LTI-REACHABILITY is ◮ undecidable if U is a finite union of affine subspaces. ◮ Skolem-hard if U = { 0 } ∪ V where V is an affine subspace ◮ Positivity-hard if U is a convex polytope Given s ∈ Q d and A ∈ Q d × d : ◮ Skolem problem : decide if ∃ T ∈ N such that ( A T s ) 1 = 0, ◮ Positivity problem : decide if ( A T s ) 1 � 0 for all T ∈ N . Why is this a hardness result? Decidability of Skolen and Positivity has been open for 70 years! 6 / 12

Our results : hardness Study the impact of the control set on the hardness of reachability Theorem LTI-REACHABILITY is ◮ undecidable if U is a finite union of affine subspaces. ◮ Skolem-hard if U = { 0 } ∪ V where V is an affine subspace ◮ Positivity-hard if U is a convex polytope Given s ∈ Q d and A ∈ Q d × d : ◮ Skolem problem : decide if ∃ T ∈ N such that ( A T s ) 1 = 0, ◮ Positivity problem : decide if ( A T s ) 1 � 0 for all T ∈ N . Why is this a hardness result? Decidability of Skolen and Positivity has been open for 70 years! Since we cannot solve Skolem/Positivity, we need some strong assumptions for decidability. 6 / 12

Our results : a positive result A LTI system ( s , A , t , U ) is simple if s = 0 and 7 / 12

Our results : a positive result A LTI system ( s , A , t , U ) is simple if s = 0 and ◮ U is a bounded polytope that contains 0 in its (relative) interior, 7 / 12