10: Differential Equations & Differential Invariants Logical Foundations of Cyber-Physical Systems André Platzer Logical Foundations of Cyber-Physical Systems André Platzer André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 1 / 19
Outline Learning Objectives 1 A Gradual Introduction to Differential Invariants 2 Global Descriptive Power of Local Differential Equations Intuition for Differential Invariants Deriving Differential Equations Differentials 3 Syntax Semantics of Differential Symbols Semantics of Differential Equations Soundness Example Proofs Soundness Proof 4 Summary 5 André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 2 / 19
Outline Learning Objectives 1 A Gradual Introduction to Differential Invariants 2 Global Descriptive Power of Local Differential Equations Intuition for Differential Invariants Deriving Differential Equations Differentials 3 Syntax Semantics of Differential Symbols Semantics of Differential Equations Soundness Example Proofs Soundness Proof 4 Summary 5 André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 2 / 19
Learning Objectives Differential Equations & Differential Invariants discrete vs. continuous analogies rigorous reasoning about ODEs induction for differential equations differential facet of logical trinity CT M&C CPS understanding continuous dynamics semantics of ODEs relate discrete+continuous operational CPS effects André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 3 / 19
Differential Facet of Logical Trinity Axiomatics Syntax Semantics Syntax defines the notation What problems are we allowed to write down? Semantics what carries meaning. What real or mathematical objects does the syntax stand for? Axiomatics internalizes semantic relations into universal syntactic transformations. How does the semantics of e = ˜ e relate to the semantics of e − ˜ e = 0, syntactically? What about derivatives? André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 4 / 19
Outline Learning Objectives 1 A Gradual Introduction to Differential Invariants 2 Global Descriptive Power of Local Differential Equations Intuition for Differential Invariants Deriving Differential Equations Differentials 3 Syntax Semantics of Differential Symbols Semantics of Differential Equations Soundness Example Proofs Soundness Proof 4 Summary 5 André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 4 / 19
ODE Examples Solutions more complicated than ODE ODE Solution x ′ = 1 , x ( 0 ) = x 0 x ( t ) = x 0 + t x ′ = 5 , x ( 0 ) = x 0 x ( t ) = x 0 + 5 t x ′ = x , x ( 0 ) = x 0 x ( t ) = x 0 e t x ′ = x 2 , x ( 0 ) = x 0 x 0 x ( t ) = x ( t ) = √ 1 − tx 0 x ′ = 1 x , x ( 0 ) = 1 1 + 2 t ... y ( x ) = e − x 2 y ′ ( x ) = − 2 xy , y ( 0 ) = 1 t 2 x ′ ( t ) = tx , x ( 0 ) = x 0 x ( t ) = x 0 e 2 x ′ = √ 4 ± t √ x 0 + x 0 x ( t ) = t 2 x , x ( 0 ) = x 0 x ′ = y , y ′ = − x , x ( 0 ) = 0 , y ( 0 ) = 1 x ( t ) = sin t , y ( t ) = cos t x ′ = 1 + x 2 , x ( 0 ) = 0 x ( t ) = tan t x ( t ) = e − 1 x ′ ( t ) = 2 t 2 non-analytic t 3 x ( t ) x ′ = x 2 + x 4 ??? x ′ ( t ) = e t 2 non-elementary André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 5 / 19
Global Descriptive Power of Local Differential Equations Descriptive power of differential equations Descriptive power: differential equations characterize continuous 1 evolution only locally by the respective directions. Simple differential equations describe complicated physical processes. 2 Complexity difference between local description and global behavior 3 Analyzing ODEs via their solutions undoes their descriptive power. 4 Let’s exploit descriptive power of ODEs for proofs! 5 x ( t ) = sin( t ) = t − t 3 3 ! + t 5 5 ! − t 7 7 ! + t 9 x ′′ = − x 9 ! − ... x ′′ ( t ) = e t 2 no elementary closed-form solution André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 6 / 19
Global Descriptive Power of Local Differential Equations You also prefer loop induction to unfolding all loop iterations, globally . . . Descriptive power of differential equations Descriptive power: differential equations characterize continuous 1 evolution only locally by the respective directions. Simple differential equations describe complicated physical processes. 2 Complexity difference between local description and global behavior 3 Analyzing ODEs via their solutions undoes their descriptive power. 4 Let’s exploit descriptive power of ODEs for proofs! 5 x ( t ) = sin( t ) = t − t 3 3 ! + t 5 5 ! − t 7 7 ! + t 9 x ′′ = − x 9 ! − ... x ′′ ( t ) = e t 2 no elementary closed-form solution André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 6 / 19
Intuition for Differential Invariants Differential Invariant Γ ⊢ F , ∆ F ⊢ ??? F F ⊢ P Γ ⊢ [ x ′ = f ( x )] P , ∆ [ ′ ] [ x ′ = f ( x )] P ↔ ∀ t ≥ 0 [ x := y ( t )] P (y’ =f(y), y(0)=x) André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 7 / 19
Intuition for Differential Invariants Differential Invariant Γ ⊢ F , ∆ F ⊢ ??? F F ⊢ P Γ ⊢ [ x ′ = f ( x )] P , ∆ [ ′ ] [ x ′ = f ( x )] P ↔ ∀ t ≥ 0 [ x := y ( t )] P (y’ =f(y), y(0)=x) André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 7 / 19
Intuition for Differential Invariants Differential Invariant Γ ⊢ F , ∆ F ⊢ ??? F F ⊢ P Γ ⊢ [ x ′ = f ( x )] P , ∆ [ ′ ] [ x ′ = f ( x )] P ↔ ∀ t ≥ 0 [ x := y ( t )] P (y’ =f(y), y(0)=x) André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 7 / 19
Intuition for Differential Invariants Differential Invariant Γ ⊢ F , ∆ F ⊢ ??? F F ⊢ P Γ ⊢ [ x ′ = f ( x )] P , ∆ Want: formula F remains true in the direction of the dynamics F ¬ F ¬ F [ ′ ] [ x ′ = f ( x )] P ↔ ∀ t ≥ 0 [ x := y ( t )] P (y’ =f(y), y(0)=x) Next step is undefined for ODEs. But don’t need to know where exactly the system evolves to. Just that it remains somewhere in F . Show: only evolves into directions in which formula F stays true. André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 7 / 19
Guiding Example v 2 + w 2 = r 2 → [ v ′ = w , w ′ = − v ] v 2 + w 2 = r 2 André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 8 / 19
Guiding Example: Rotational Dynamics v 2 + w 2 = r 2 → [ v ′ = w , w ′ = − v ] v 2 + w 2 = r 2 w v r sin ϑ w = r cos ϑ r v André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 8 / 19
Guiding Example: Rotational Dynamics v 2 + w 2 = r 2 → [ v ′ = w , w ′ = − v ] v 2 + w 2 = r 2 ⊢ v 2 + w 2 − r 2 = 0 → [ v ′ = w , w ′ = − v ] v 2 + w 2 − r 2 = 0 → R André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 8 / 19
Outline Learning Objectives 1 A Gradual Introduction to Differential Invariants 2 Global Descriptive Power of Local Differential Equations Intuition for Differential Invariants Deriving Differential Equations Differentials 3 Syntax Semantics of Differential Symbols Semantics of Differential Equations Soundness Example Proofs Soundness Proof 4 Summary 5 André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 8 / 19
Syntax With Primes e ::= x | c | e + k | e − k | e · k | e / k Syntax André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 9 / 19
Syntax With Primes e ::= x | c | e + k | e − k | e · k | e / k Syntax ( e + k ) ′ = ( e ) ′ +( k ) ′ ( e − k ) ′ = ( e ) ′ − ( k ) ′ ( e · k ) ′ = ( e ) ′ · k + e · ( k ) ′ Derivatives ( e / k ) ′ = ( e ) ′ · k − e · ( k ) ′ � � / k 2 ( c ()) ′ = 0 for constants/numbers c () André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 9 / 19
Syntax With Primes e ::= x | c | e + k | e − k | e · k | e / k Syntax ( e + k ) ′ = ( e ) ′ +( k ) ′ ( e − k ) ′ = ( e ) ′ − ( k ) ′ ( e · k ) ′ = ( e ) ′ · k + e · ( k ) ′ Derivatives ( e / k ) ′ = ( e ) ′ · k − e · ( k ) ′ � � / k 2 same singularities ( c ()) ′ = 0 for constants/numbers c () André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 9 / 19
Syntax With Primes e ::= x | c | e + k | e − k | e · k | e / k Syntax ( e + k ) ′ = ( e ) ′ +( k ) ′ ( e − k ) ′ = ( e ) ′ − ( k ) ′ ( e · k ) ′ = ( e ) ′ · k + e · ( k ) ′ Derivatives ( e / k ) ′ = ( e ) ′ · k − e · ( k ) ′ � � / k 2 same singularities ( c ()) ′ = 0 for constants/numbers c () . . . What do these primes mean? . . . André Platzer (CMU) LFCPS/10: Differential Equations & Differential Invariants LFCPS/10 9 / 19
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